Time Evolution of Quantum Resonance States Dissertation zur Erlangung des akademischen Grades ”Doctor rerum naturalium” (Dr. rer. nat.) genehmigte Dissertation eingereicht am Institut f¨ ur Mathematik Fakult¨atII der Technischen Universit¨at Berlin von Dipl.-Phys. Juliane Rama aus Berlin Promotionsausschuß: Vorsitzender: Prof. Dr. M. Scheutzow, Institut f¨ ur Mathematik, Technische Universit¨ at Berlin Gutachter: Prof. Dr. R. W¨ ust, Institut f¨ ur Mathematik, Technische Universit¨ at Berlin Gutachter: Prof. Dr. M. Klein, Institut f¨ ur Mathematik, Universit¨ at Potsdam Tag der wissenschaftlichen Aussprache: 27. November 2007 Berlin 2007 D 83
Transcript
Time Evolution of Quantum Resonance
States
Dissertation zur Erlangung des akademischen Grades”Doctor rerum naturalium” (Dr. rer. nat.)
genehmigte Dissertation
eingereicht amInstitut fur Mathematik
Fakultat IIder Technischen Universitat Berlin
von
Dipl.-Phys. Juliane Ramaaus Berlin
Promotionsausschuß:
Vorsitzender: Prof. Dr. M. Scheutzow, Institut fur Mathematik,Technische Universitat Berlin
Gutachter: Prof. Dr. R. Wust, Institut fur Mathematik,Technische Universitat Berlin
Gutachter: Prof. Dr. M. Klein, Institut fur Mathematik,Universitat Potsdam
Tag der wissenschaftlichen Aussprache: 27. November 2007
Berlin 2007
D 83
Deutsche Zusammenfassung der Dissertation
Bezeichne H := H0 + Storung den Hamiltonoperator (selbstadjungiert in einem Hilbert-raum) eines quantenmechanischen Systems. Sehr viele Resonanzphanomene in der Quan-tenmechanik hangen mit ”kleinen” Storungen von eingebetteten Eigenwerten des (ebenfallsselbstadjungierten) ungestorten Hamiltonoperators H0 zusammen. Resonanzzustande wer-den mit metastabilen Zustanden identifiziert. Metastabile Zustande sind Zustande, die – bisauf einen (moglicherweise zeitabhangigen) Restterm – exponentielles zeitliches Abfallverhal-ten zeigen.
Sei λ0 ein eingebetteter Eigenwert von H0, H0ψ0 = λ0ψ0. Dann ist zu erwarten, daßder ”eingebettete” Eigenzustand ψ0 unter Storung metastabiles Verhalten zeigt. Ziel dieserDissertation ist, im Falle eines solchen eingebetteten Eigenwertes λ0 beliebiger Vielfachheitfur eine moglichst allgemeine Klasse von Storungen eine direkte dynamische Interpretationdes ”naiven” Resonanzzustands e−itHψ0 zu liefern. Resultate dieser Art existieren bisherausschließlich im Falle nicht entarteter (also einfacher) eingebetteter Eigenwerte λ0; siehez.B. [Hu2], [CGrHu].
Part 1. Motiviert durch Ergebnisse aus [MeSi], formuliert in einem abstrakten Hilbertraum-Setting mit Hilfe eines Operatorkalkuls basierend auf Mourre-Abschatzungen, wurden fol-gende Resultate erzielt:
SeiH0 ein selbstadjungierter (ungestorter) Hamiltonoperator in einem komplexen Hilber-traum 〈H, ‖ · ‖〉. Sei λ0 ein Eigenwert beliebiger Vielfachheit von H0, eingebettet in seinstetiges Spektrum σc(H0) mit (normierter) Eigenfunktion ψ0: H0ψ0 = λ0ψ0. Sei Π0 Or-thogonalprojektor auf Ker(H0−λ0), dimRanΠ0 ≤ ∞ und Π0 := 1 −Π0. Fur eine abstrakteKlasse von symmetrischen Storungen W und H := H0 + W wird die Asymptotik des(naiven) Resonanzzustands e−itHψ0 im limes W → 0 entwickelt. Es wird gezeigt, daße−itHψ0 das typische Verhalten eines Resonanzzustands zeigt: Bis zur Großenordnung derzu erwartenden Lebensdauer des Zustands – gegeben durch Fermis Goldene Regel – zeigt‖Π0e
−itHψ0‖ im wesentlichen exponentielles Verhalten; vgl. (1.24). Fur große Zeiten hinge-gen kann e−itHψ0 vollstandig in das Spektralkomplement RanΠ0 tunneln, wo der Zustandin einem schwachen Sinn auslaufend (outgoing) ist: e−itHψ0 gehort zu großen Spektral-werten eines (geeigneten) Operators A konjugiert zu H (im Sinn von (C1)-(C4)). Das heißt,die gewichtete Norm ‖〈A〉−αΠ0e
−itHψ0‖ (α > 2 geeignet) ist uniform klein in der Zeit imSinne von (1.26).
Part 2 ist motiviert durch die Frage, ob (unter starkeren Voraussetzungen an die Klassevon Storungen) eine Verbesserung der Zeitkontrolle fur Zeiten uber die erwartete Lebens-dauer hinaus moglich ist.
Part 2. Angeregt durch Hunzikers Arbeit [Hu2] wurden folgende Ergebnisse erzielt:
Bezeichne H(κ) = H0 + κV (κ ≥ 0 klein genug) eine Familie selbstadjungierterOperatoren im Hilbertraum eines abstrakt-dilatationsanalytischen Systems (im Sinne von(A1) und (A5)). Die Familie H(κ) sei außerdem analytisch im verallgemeinerten Sinn (sieheDefinition F.2) in der Variablen κ fur κ in einer komplexen Nullumgebung. Sei λ0 ein einge-betteter Eigenwert endlicher Vielfachheit von H0. Bezeichne Π0 den zugehorigen Eigen-projektor, dim RanΠ0 < ∞. Fur den speziellen Fall dimRanΠ0 = 2 wird die Asymptotikvon ‖Π0e
−itH(κ)Π0‖ im limes κ → 0 analysiert. Einige der Resultate bleiben auch im Falle
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beliebiger Dimension dim RanΠ0 <∞ bestehen. Dimensionsunabhangig gilt auf RanΠ0 dieOperatorrelation
Π0e−itH(κ)Π0 = e−ith(κ) +Rest(κ, t) (κ ≥ 0 klein genug, t ≥ 0) ; (0.1)
vgl. (2.172). Dabei ist der effektive Hamiltonoperator h(κ) eine analytische Familie nichtselbstadjungierter Endomorphismen auf RanΠ0. Eigenwerte von h(κ) sind nach Konstruk-tion komplex und stimmen mit den Resonanzeigenwerte, die unter Storung aus λ0 hervorge-hen, uberein. Fur dim RanΠ0 = 2 zeigt ‖Π0e
−itH(κ)Π0‖ grob exponentielles Abfallverhalten(bis auf einen zeitabhangigen Restterm; siehe Theorem 2.41). Dieses exponentielle Verhal-
ten ist bestimmt durch einen Faktor e−κ2γt, wobei γ eine echt positive Konstante darstellt,die durch die Imaginarteile der Resonanzeigenwerte bestimmt wird; siehe (2.175), (2.176)und Proposition 2.40. Der zeitabhangige Restterm wird fur Zeiten 0 ≤ t = O((− lnκ)κ−2)(κ → 0) durch den exponentiellen Anteil dominiert; vgl. Corollary 2.42. Dies ist eine log-arithmische Verbesserung der Zeitkontrolle (uber die erwartete mittlere Lebensdauer ∼ κ−2
hinaus), verglichen mit den Ergebnissen aus Part 1, Theorem 1.8. Fur festes κ > 0 kleingenug gilt limt→∞ ‖Π0e
−itH(κ)Π0‖ = O(κ2). Die logarithmische Verbesserung der Zeitkon-trolle uber den exponentiellen Hauptterm ist das zentrale Ergebnis in Part 2.
Abschließend ist zu bemerken, daß die Entwicklung der Asymptotik von ‖Π0e−itH(κ)Π0‖
im limes κ→ 0 im Falle beliebiger Dimension dim RanΠ0 <∞ ein kombinatorisches Problemdarstellt (das in Bearbeitung ist).
Ferner setzt die Anwendung der in Part 2.2 entwickelten Methoden (die auf einerJordanzerlegung von h(κ) beruhen) nicht zwingend Dilatationsanalytizitat des quanten-mechanischen Modells voraus: Wann immer eine endlichdimensionale Reduktion der Art(0.1) gegeben ist (sich also die gestorte Dynamik als Summe aus Exponentialfunktion eineranalytischen Matrixfamilie und zeitabhangigem Restterm schreiben laßt), sind die Methodenaus Part 2.2 prinzipiell anwendbar.
Danksagung
Ich bedanke mich herzlich bei meinen beiden Doktorvatern Rainer Wust und Markus Kleinfur die gute Zusammenarbeit und viele hilfreiche Diskussionen. Es war mir eine Freude undeine Ehre!
In quantum mechanics most resonance phenomena are related to perturbations of embeddedeigenvalues for an unperturbed (self-adjoint) Hamiltonian H0, if
H = H0 + perturbation
describes the self-adjoint Hamiltonian of the quantum system in question. For an appro-priate small perturbation (where ”small” has to be specified from case to case) an embed-ded eigenvalue of H0 should turn into a resonance; see, e.g., [AHerSk], [Sim3]. Quantumresonance states are then identified with metastable states. Metastable states typically showexponential decay for times approximate up to the order of their expected lifetime given bythe Fermi golden rule (a perturbation formula for embedded eigenvalues). Since the lastfew decades different mathematical concepts (as well stationary as explicit time-dependentconcepts) have been developed to describe metastable states in a rigorous way. A rigorousmathematical description involves quite a few problems; see [Sim4] for an overview.In this thesis we derive a direct dynamical interpretation in Hilbert space of eigenstates cor-responding to degenerate embedded eigenvalues of H0 under the perturbed dynamics e−itH
for certain (little restrictive) classes of perturbations. Therefore we use a time-dependentapproach in Part 1 and a stationary one in Part 2. In Part 3 we discuss our results.
We briefly recall some approaches to quantum resonances which have been developed inthe past and contextualize our new results:
Stationary methods (”analytic deformations”). The basic ideas of complex scaling goback to Aguilar, Balslev, Combes and were first applied in [AgCo] and [BCo] to investigatespectral analysis of quantum (n-body) systems described by a Hamiltonian of the formH = T0 + V , where T0 denotes the kinetic energy of the system and V is a (sum of)dilation analytic two-body interaction(s) in the class Cβ (cf. Definition F.7). The meritsof [AgCo] and [BCo] are proving the absence of continuous singular part for the spectrumσ(H) and proving that eigenspaces of H associated with non-threshold eigenvalues are finitedimensional ([BCo, Theorem 1], [AgCo, Theorem III.1]).
It was Simon [Sim3] who applied dilation analyticity for studying embedded eigen-values (of some unperturbed Hamiltonian like H0 = T0 + V ) and resonances of non-relativistic many-body quantum systems with dilation analytic interactions, including two-body Coulomb and Yukawa interactions. (A reduction due to symmetry of the system inquestion is used in [Sim3].): If H = H0 + perturbation denotes the perturbed Hamil-tonian of a dilation analytic system (the perturbation describes, e.g., the electron-electronrepulsion), being self-adjoint in L2(Rν), then by use of the dilation group U(θ) (cf. (F.2))H is dilated to an analytic family H(θ)θ∈S for θ in some complex strip S ⊂ C. Thenon-real eigenvalues of such a dilated Hamiltonian H(θ) are defined to be the resonancesof the system. Under perturbation and dilation embedded eigenvalues of H0 should turninto resonances; see, e.g., [Sim3, Theorem 3.2]. By Balslev-Combes theory (non-threshold)
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embedded eigenvalues of H0 are real isolated eigenvalues of the dilated operator H0(θ) fornon-real dilation parameter θ; see [BCo, Lemma 3]. Thus Rayleigh-Schrodinger perturba-tion theory for discrete eigenvalues can be applied. (This is in fact the key tool to show thatembedded eigenvalues perturb to resonances.) Furthermore, Simon [Sim3] derived an exactmathematical formula which heuristically is equal to the physics text books versions of theFermi golden rule (which mostly are rather imprecisely defined); see [Sim3, Theorem 4.1].
Further, Simon [Sim2] introduced a form analogue Fβ (cf. Definition F.8) of theclass Cβ .
Moreover, there exist some other generalizations and extensions of complex scaling toanalyze physical situations and systems where resonances are expected but the conditionsof complex scaling in its original setting are not fulfilled. Examples are: Exterior scaling[Sim5] and modified analytic distortions [Cy], [Hu1]. Both methods allow the analysis ofBorn-Oppenheimer approximation; see, e.g., [Kl], [KlMSW].
We further remark that Agmon [A] developed a stationary resonance theory for closeddensely defined linear operators in Banach spaces, where resonances are identified withcertain poles of a generalized resolvent; cf. [A, Definition 3.1]. Agmon’s paper does notaddress time evolution of resonance vectors (defined in [A, Definition 5.4]).
All these stationary approaches based on ”analytic deformations” require extra argu-ments to describe the time behavior. For results in this direction we mention, e.g., [Hu2] for(abstract) dilation analytic systems or in a semiclassical context [GSi].
Stationary methods and time behavior. We shall now sketch the results of [Hu2]:For self-adjoint Hamiltonians H(κ) = H0 + κV , κ ≥ 0 small enough, satisfying an
abstract Balslev-Combes setting, Hunziker [Hu2] treats resonance states which appear asperturbed bound states. The corresponding metastable states are constructed using a (for-mal) Rayleigh-Schrodinger expansion to order N − 1 for the (nonexistent) perturbed boundstate. This setting applies in arbitrary order N ≥ 1 to cases like the stark effect and inlowest order N = 1 to perturbation of bound states embedded in the continuum. We shallconcentrate on the latter case. So let λ0 be an embedded eigenvalue of H0 with corre-sponding eigenprojection Π0, H0ψ0 = λ0ψ0, ψ0 normalized. For perturbation of a simpleembedded eigenvalue λ0 the result of [Hu2] gives an explicit description of the physical in-tuition, which is a roughly exponential decay of the ”naive” resonance state e−itH(κ)ψ0:(ψ0, e
−itH(κ)ψ0) = e−itλ(κ) + O(κ2) (κ → 0) uniformly in t ≥ 0, where λ(κ) denotes theresonance (eigenvalue) in the sense described above; cf. [Hu2, (35)]). For λ0 a (finitely)degenerate embedded eigenvalue of H0 the perturbed dynamics for states in RanΠ0 are for-mulated as an operator relation on RanΠ0: Let ∆ be a small enough interval about λ0 andlet g∆ denote a smoothed out characteristic function. Then for t ≥ 0 the perturbed dynam-ics e−itH(κ)g∆(H(κ)) are approximately given (in the sense of [Hu2, Theorem 2, (36)]) bythe dynamics e−ith(κ) of an effective non self-adjoint Hamiltonian h(κ) (which is an endo-morphism on RanΠ0 and thus finite-dimensional) up to a time-dependent remainder. Thisresult finally leads to (see [Hu2, (41)])
(t(κ)ϕ, e−itH(κ)t(κ)ψ
)=
(ϕ, e−ith(κ)ψ
)+O(κ2) (κ→ 0) (0.1)
uniformly for states ϕ, ψ ∈ RanΠ0 and t ≥ 0, where t(κ)ψ describes the (formal) perturbedbound state of order 1 − 1 = 0. The proof is based on Stone’s formula. The asymptoticsof the main term e−ith(κ) have not been worked in [Hu2]. As follow-up papers of [Hu2] oneshould mention [CGrHu] and [JNe].
In [CGrHu] for a simple embedded eigenvalue of the unperturbed Hamiltonian the resultof [Hu2] has been generalized to quantum systems with Hamiltonians H(κ) = H0 + κV
characterized by Mourre estimates and smoothness of the resolvent instead of dilation ana-lyticity.
In [JNe] a family of self-adjoint Hamiltonians H(κ) = H0 +κV for κ small is considered,where no assumptions on (dilation) analyticity have been made. If λ0 denotes a finitelydegenerate eigenvalue of H0 and Π0 its corresponding eigenprojection, Jensen and Nenciu
Time Evolution of Quantum Resonance States 3
derived in [JNe, Theorem 4] – in analogy to [Hu2] – an effective Hamiltonian h(κ) on RanΠ0
and an error term δ(κ, t) such that
Π0e−itH(κ)Π0 = e−ith(κ) + δ(κ, t) , sup
t≥0‖δ(κ, t)‖ ≤ cκ2 (0.2)
for t ≥ 0 and some c ≥ 0. This result hold under quite technical assumptions on somereduced resolvent G(z) of H0 (cf. [JNe, (1.9) and (2.9) ff.]) and the usual Fermi golden rulecondition for embedded eigenvalues [JNe, (2.21)].
Time dependent methods. The generalized framework to describe local decay estimatesof quantum resonance states uses the structure of weighted Hilbert spaces. Local decayestimates are formulated with the help of some self-adjoint operator A in a (complex) Hilbertspace H, conjugate to the Hamiltonian H of the system in question. (This conjugation hasto be specified in each case.) The operator weight 〈A〉α := (|A|2 + 1)α/2 for some α > 0large enough measures the localization of states ψ in H in the following sense: Functionsψ ∈ H in the domain of 〈A〉α can be called well-localized (in the spectral representation ofA). For ψ(t, ·) := e−itHψ(0, ·) the rate of convergence for
‖〈A〉−αψ(t, ·)‖H → 0 (t→ ∞) (0.3)
is defined to be the local decay estimate for ψ(t, ·). Therefore the state ψ(t, ·) belongs to asubspace of large spectral values of A and is outgoing in the sense of (0.3). First results inthis direction have been obtained until a few years ago. They all go back to a publicationof A. Orth, [O]. We mention, e.g., [SoWei], [CosSo], [MeSi].
By use of weighted L2-spaces Soffer and Weinstein [SoWei] presented a time-dependenttheory of quantum resonances occurring as perturbations of systems having simple embeddedeigenvalues in their continuous spectrum. Among some rather technical assumptions thekey hypotheses are a ”non-vanishing of the Fermi golden rule” (the resonance condition; cf.[SoWei, (W3)]) and a local decay estimate for the unperturbed dynamics (see [SoWei, (H4)],where the operator A in (0.3) has been chosen as multiplication by the coordinate x) withinitial data consisting of continuum modes associated with an interval containing the simpleembedded eigenvalue of the unperturbed Hamiltonian; see [SoWei, (H4)]. The main results[SoWei, Theorem 2.1, Corollary 2.1] can be summarized as follows: Under quite generalassumptions on the unperturbed Hamiltonian H0 and the symmetric perturbation (which isassumed to be small) the perturbed Hamiltonian H has absolutely continuous spectrum inan interval about a simple embedded eigenvalue λ0 of H0. If ψ0 denotes the (normalized)eigenfunction corresponding to λ0, solutions of i∂tψ = Hψ with initial condition ψ0 arecharacterized by transient exponential decay. The exponential rate (reciprocal of lifetime)can be calculated.
Ideas of [SoWei] have been crucial for the time-dependent approach to quantum reso-nances of [MeSi]. For a bigger class of symmetric perturbations and degenerate embeddedeigenvalues λ0 of H0 the dynamics of ”resonance states” are analyzed in an abstract Hilbertspace setting, using an operator calculus based on Mourre estimates. Therefore let A bean auxiliary self-adjoint operator in Hilbert space, conjugate to H = H0 + W . Let λ0 bea possible degenerate eigenvalue of the unperturbed Hamiltonian H0 with correspondingeigenprojection Π0, dimRanΠ0 ≤ ∞. The smallness of the perturbation is measured bythe parameter κ := ‖〈A〉αWΠ0‖ < ∞ for some α > 2. Let E∆(H) denote the spectralprojection with respect to H onto some small (enough) interval ∆ about λ0. Then solu-tions ψ(t) = e−itHψ(0) of i∂tψ = Hψ with initial conditions ψ(0) ∈ E∆(H) ∩ D(〈A〉α) canbe interpreted as resonance states. ψ(t) is the sum of a dispersive (locally decaying) partψdisp(t) and a ”resonance part” ψres(t), which reveals the quasi exponential decay of ψ(t):ψ(t) = ψres(t) + ψdisp(t), where for some α > 2
‖〈A〉−αψdisp(t)‖ → 0 (t→ ∞) ;
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cf. [MeSi, (2.5)], respectively (1.12) in this thesis. ψres(t) is authoritatively determined bythe projection onto RanΠ0:
ψres(t) =(1 +O(κ)
)Π0ψ(t) (κ→ 0, t ≥ 0) .
With some complex bounded operator Λ and some β > 0 it holds
Π0ψ(t) = e−iΛtΠ0ψ(0) +O(κ1−4β(t+ 1)−β
)(κ→ 0 , t ≥ 0) ,
Re Λ = λ0 +O(κ) , ImΛ = −Γ +O(κ3) (κ→ 0 , t ≥ 0) ;
see [MeSi, Theorem 2.1], respectively Theorem 1.3 in this thesis. The bounded operatorΓ = O(κ2) (κ → 0) is an operator version of the Fermi golden rule. Γ is assumed to bestrictly positive on RanΠ0. The positivity of Γ on RanΠ0 among a local decay estimate for
e−itH with H := (1 −Π0)H(1 −Π0) (cf. [MeSi, (2.1)]) are the main assumptions of [MeSi].We remark that our results of Part 1 are based on the results of [MeSi].
Our results. Assume λ0 being a degenerate embedded eigenvalue (of arbitrary multiplicity)of an unperturbed Hamiltonian H0 of the quantum system in question with correspondingeigenprojection Π0, dim RanΠ0 ≤ ∞. We address the asymptotics of quantum resonancestates resulting from perturbation of such a degenerate embedded eigenvalue of H0. To thebest of our knowledge in the literature results in this direction do only exist in form of opera-tor relations as for example given in [Hu2, Theorem 2], [MeSi, Theorem 2.1], [JNe, Theorem4]. Our goal is to give estimates for the perturbed dynamics in this multidimensional prob-lem being of that quality in the one-dimensional case (corresponding to a simple eigenvalue);compare with, e.g., [Hu2, Theorem 1, (35)] or [CGrHu, Theorem 1.2]. Roughly speakingwe want to derive a direct dynamical interpretation of resonance states corresponding toperturbations of degenerate embedded eigenvalues.
In Part 1 of this thesis we amplify the results of Merkli and Sigal [MeSi]. We assumeH0 to be self-adjoint in a complex Hilbert space 〈H, ‖ · ‖〉 with λ0 a degenerate eigenvalueof arbitrary multiplicity embedded in the continuous spectrum σc(H0) and ψ0 the corre-sponding normalized eigenfunction ψ0, H0ψ0 = λ0ψ0. Let Π0 denote the eigenprojectioncorresponding to λ0, dim RanΠ0 ≤ ∞. Define Π0 := 1 − Π0. For a certain class of sym-metric perturbations W (the same class of perturbations considered in [MeSi], much moregeneral than the class of dilation analytic perturbations) and H = H0 +W we investigatethe asymptotics of the ”naive” resonance state e−itHψ0 in the limit W → 0. We prove thate−itHψ0 shows the typical behavior of a resonance state: Up to the order of the expectedlifetime (given by the Fermi golden rule) ‖Π0e
−itHψ0‖ decays roughly exponentially; cf.(1.24). For large times, however, e−itHψ0 may tunnel completely to the spectral comple-ment RanΠ0. But there it is outgoing in the sense of (1.26). In Theorem 1.8 we give a directdynamical interpretation of such resonance states independent of the degeneracy of the em-bedded eigenvalue λ0. In the literature we did not found any results on direct dynamicalinterpretation of resonances states resulting from a degenerate embedded eigenvalue of arbi-trary multiplicity. The fact that in (1.24) we cannot control the main term (i.e., ‖e−Γtψ0‖)for times exceeding the lifetime ∼ κ−2 is due to the additive error term O(κ2t). But as wewill see in the Discussion (Part 3), without further assumptions on the analytic structure ofthe Hamiltonian H our estimates in Theorem 1.7 and Theorem 1.8 seem to be near optimal.
Part 2 of this thesis is motivated by the question whether one can control the main termin such asymptotics for times larger than the expected lifetime, of course under strongerconditions on the considered Hamiltonian:
In Part 2 we consider dilation analytic perturbations V and Hamiltonians H(κ) =H0+κV self-adjoint in a complex Hilbert space for κ ≥ 0 small enough, H(κ) being analyticin the generalized sense in the variable κ in some complex neighborhood of κ = 0. We ana-lyze the asymptotics Π0e
−itH(κ)Π0 in the limit κ → 0 for the special case dimRanΠ0 = 2.Therefore we used the results of [Hu2]. Some of our results also persist in case of ar-bitrary dim RanΠ0 < ∞; see, e.g., (2.123), Remark 2.34. The general case of arbitrary
Time Evolution of Quantum Resonance States 5
dimRanΠ0 < ∞ is a combinatorial problem which is still in preparation. In analogy to[Hu2, Theorem 2] independent of dim RanΠ0 we get the operator relation on RanΠ0
(see (2.172)), where h(κ) is some (not self-adjoint) analytic matrix family. FordimRanΠ0 = 2 it turns out that ‖Π0e
−itH(κ)Π0‖ decays – up to a time-dependent re-
mainder – roughly exponentially, driven by a term e−κ2γt (see Theorem 2.41), where γ issome strictly positive constant determined by the imaginary parts of the resonance eigen-values (cf. Proposition 2.40). By construction the resonance eigenvalues coincide with theeigenvalues of h(κ). The key tool to obtain these results is a Jordan decomposition of theanalytic matrix family h(κ).
According to Rayleigh-Schrodinger perturbation theory the expected lifetime is of orderO(κ−2) as κ→ 0. For times 0 ≤ t = O((− lnκ)κ−2) (κ→ 0) the time-dependent remainderis dominated by the exponential term; see Corollary 2.42. This is a logarithmic improvementof time control with regard to the results of Theorem 1.8. This may seem weak, but reallyneeds a lot of analytic structure and is in fact the central result. This better time control iscaused by the multiplicative character of the (critical) error terms in (2.175) and (2.176).
Although motivated by the dilation analytic setting of [Hu2], dilation analyticity is notcrucial for our methods and results of Part 2. Important is the existence and analyticityof an effective Hamiltonian h(κ): Assume H(κ) = H0 + κV (κ ≥ 0 small) to be a self-adjoint Hamiltonian in a complex Hilbert space, V a symmetric perturbation (without anyassumptions on dilation analyticity) and λ0 a finitely degenerate eigenvalue of H0 withcorresponding eigenprojection Π0. Whenever one has a ”finite-dimensional reduction” toRanΠ0 of the form
where the effective (not self-adjoint) Hamiltonian h(κ) is an analytic matrix family on RanΠ0
for κ in some complex neighborhood of κ = 0, then our methods are applicable to this setting;they should yield results very similar to our results of Theorem 2.41 and Corollary 2.42, ifthe remainder suffices r(κ, t) = O(κp) for some p > 0 (large enough) as κ → 0, uniformlyin t ≥ 0. In particular our methods seem to be applicable to (a subclass of) HamiltoniansH(κ) satisfying the conditions of [JNe, Theorem 4]. This should yield the same time controlas given in Corollary 2.42, since in [JNe, Theorem 4] the remainder (roughly comparableto the remainder B(κ, t) in our setting of Theorem 2.41) also is of order O(κ2) as κ → 0,uniformly in t ≥ 0; cf. (0.2). Unfortunately we did not notice the preprint [JNe] until anadvanced stage of working out Part 2.
Part 1
Time Evolution of Quantum Resonance States I:A Time-Dependent Theory
1.1 Introduction and Results
Let H0 be a self-adjoint Hamiltonian in a complex Hilbert space 〈H, ‖ · ‖〉. Let λ0 be apossibly degenerate eigenvalue of H0, embedded in its continuous spectrum σc(H0), with(normalized) eigenfunction ψ0: H0ψ0 = λ0ψ0. Let Π0 be the orthogonal projection ontoKer(H0 − λ0) with dim Ran Π0 ≤ ∞ and
Π0 := 1 − Π0 . (1.1)
Now let H0 be perturbed by an operator W , where
(C0) W is symmetric in H, H := H0 +W is self-adjoint in H and D(H0) = D(H) .
For a small perturbation W – where ”small” is specified by (C2) through (C5) below – λ0
should turn into a resonance; see [AHerSk] for results in this direction. More naively, onemay directly investigate e−itHψ0. One expects that e−itHψ0 shows the typical behaviorof a resonance: Up to the order of the expected lifetime (given by the Fermi golden rule)‖Π0e
−itHψ0‖ decays (roughly) exponentially; see (1.24). For large times e−itHψ0 may tunnelcompletely to the spectral complement Ran Π0, but there it is (in some weak sense) outgoing.This can be defined to mean that e−itHψ0 belongs to a subspace of large spectral valuesfor an operator A conjugate to H (as specified below). Thus the last statement may berephrased by saying that the weighted norm ‖〈A〉−αΠ0e
−itHψ0‖ is small uniformly in time.Such an approach was introduced in [MeSi], following [SoWei]. It is the main purpose
of this thesis’ Part 1 to show that the statements above - although they are not explicitlyproved in [MeSi] - actually follow from the estimates of [MeSi] by standard techniques.
After Part 1 of this thesis was completed, we learnt of the paper [CGrHu] which treatsthe case dim Ran Π0 = 1 (under conditions on the potential roughly comparable to ours) bya stationary method following the analysis of the dilation analytic case in [Hu2]. We remarkthat the explicit asymptotics for the main term in the degenerate case dimRan Π0 > 1 hasnot been worked out in [Hu2].
In Part 2 of this thesis we will investigate these explicit asymptotics in a dilation analyticsetting closely related to [Hu2] for the degenerate case dim RanΠ0 = 2 by using an appro-priate Jordan decomposition. This shall improve our remainder estimates of this section.We think that the case dim RanΠ0 = N for arbitrary N < ∞ can be treated by the samemethods (based on an appropriate Jordan decomposition) as used in case N = 2. But thisturns out to be a hard combinatorial problem, which is in preparation.
We remark that there are related results on the time decay of resonance states in thesemiclassical case where there are resonances close to the real axis. We mention, e.g., [GSi]for the case of a simple resonance and [NStZ] for the case of many resonances at highenergies.
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8 J. Rama
To formulate our results more precisely, we shall introduce some notations and brieflyrecall the central result of [MeSi]. For any bounded interval I let gI ∈ C∞
0 be a smoothedout version of the characteristic function 1I , i.e.
gI(µ) =
1 , µ ∈ I
0 , µ outside some neighborhood of I. (1.2)
We fix some neighborhood ∆ of λ0 (assumed to contain no eigenvalue of H0 different fromλ0) and an interval ∆ ⊂ ∆′ a little bigger than ∆. g∆(H) is a smoothed out version of thespectral projection E∆(H) = 1∆(H). We assume that supp g∆ ∩ supp (1 − g∆′) = ∅ and ∆′
also contains no eigenvalues of H0 different from λ0. We set
gI := 1 − gI (1.3)
for any interval I and
H := Π0HΠ0 . (1.4)
Assume that there exists a self-adjoint operator A in H and α > 2 such that
(C1) ‖〈A〉αΠ0‖ <∞, 〈A〉 := (|A|2 + 1)1/2 .
Next we state, in addition to (C0), further conditions on W .
(C2) κ := ‖〈A〉αWΠ0‖ <∞ .
Remark 1.1 κ is a measure for the size of the perturbation W . In this work we areinterested in κ small.
(C3) The k-fold commutators adkA(H), recursively defined by adA( · ) := [A, · ], are H-
bounded for all k ∈ 1, 2, . . . , n and some n > α + 1 > 3, uniformly in κ < κ0 forsome κ0 sufficiently small.
(C4) For all φ ∈ D(〈A〉α) and t ≥ 0 the following local decay estimate holds:
‖〈A〉−αe−itHg∆′(H)Π0φ‖ ≤ C 〈t〉−α‖〈A〉αΠ0φ‖
for some C <∞, independent of t and κ < κ0 for some κ0 small enough.
〈t〉 := (1 + |t|2)1/2 (t ∈ R) ; Π0, H, ∆′, g∆′ are defined in (1.1) - (1.4).
(C5) Non-vanishing of the Fermi golden rule holds, i.e.
for some c0 > 0, uniformly in κ < κ0 for some κ0 sufficiently small.
Remark 1.2 In analogy to the well known formula
limε↓0
(x− iε)−1 = P.V.
(1
x
)
+ i · πδ(x) ,
which holds in the space of tempered distributions, we define
〈A〉−αδ(H − λ0)Π0〈A〉−α :=1
π· Im
(
s- limε↓0
〈A〉−α(H − λ0 − iε)−1Π0〈A〉−α)
, (1.6)
〈A〉−αP.V.(H − λ0)−1Π0〈A〉−α := Re
(
s- limε↓0
〈A〉−α(H − λ0 − iε)−1Π0〈A〉−α)
, (1.7)
whenever the limits on the r.h.s. exist. In fact the existence of the limits followsfrom (C4) (see Appendix A). Obviously Γ ≥ 0. The actual assumption in (C5) is thepositivity of Γ on Ran Π0. Note that Γ = O(κ2) by (C2), if the limit in (1.6) exists.
Time Evolution of Quantum Resonance States 9
We refer to [MeSi, Section 2.3, Section 4.2] for an application of this abstract setting to thecase of Schrodinger operators and a relation of (C4) to the Mourre estimate.
So the class of perturbations in question is
Wκ0:= W |W satisfies (C0) – (C5) for κ < κ0 .
Assuming (C0) - (C5), results about time evolution of resonance states have been provedin [MeSi, Theorem 2.1]. These results are formulated in terms of the bounded operator[MeSi, p.559/560 and (A.17)]
uniformly in W ∈ Wκ0. (Λ and B are defined in (1.8) and (1.9).)
Remark 1.4 Under the conditions outlined in this section, H has no eigenvalues in∆ ( cf. [MeSi, Corollary 2.2] ).
To understand the action of e−iΛt in more detail, one needs a suitable expansion of e−iΛt.As a preparation, we collect results of [MeSi, Proposition 3.3 ] and [MeSi, A. Appendix,p.573 ff.]:
Proposition 1.5 [MeSi, cp. Proposition 3.3 ]
Λ has the representation
Λ = λ0Π0 + Π0WΠ0 − Π0W(P.V.(H − λ0)
−1)Π0WΠ0 − iΓ +K , (1.13)
where
K = O(κ3) (κ→ 0) , (1.14)
Γ = O(κ2) (κ→ 0) , (1.15)
Π0W(P.V.(H − λ0)
−1)Π0WΠ0 = O(κ2) (κ→ 0) ,
uniformly in W ∈ Wκ0for some κ0 sufficiently small.
10 J. Rama
Remark 1.6 Explicitly,
K := Π0Wg∆′(H)Π0
∫
C
(H − z)−1WΠ0 Π0W (H − z)−1 1
2π
(∂g∆
)(z) dx dyΠ0 +
+ Π0W
∞∑
j=2
(
Π0g∆′(H) g∆(H))j
Π0 , (1.16)
where g∆ is an almost analytic extension of g∆ in the sense of Lemma 1.12. Our proofdoes not need this explicit representation of K.
Setting
G := λ0Π0 + Π0WΠ0 − Π0W(P.V.(H − λ0)
−1)Π0WΠ0 , Q := G− iΓ , (1.17)
we have by (1.13) Λ = Q+K.
Now we are ready to describe in more detail the asymptotic behavior of Π0ψ(t) (i.e. thedecay of the resonance state), valid up to the expected lifetime, which is O(κ−2). Thefollowing theorems are the main result of our paper.
Theorem 1.7 Assume (C0) - (C5). Let ψ(t) = e−iHtψ(0) with ψ(0) ∈ Ran(E∆(H)) ∩D(〈A〉α). Then there exists ǫ ∈ (0, 1] and C > 0 such that for 0 ≤ t ≤ Cκ−2
for some c0 > 0, c > 0, uniformly in W ∈ Wκ0for some κ0 sufficiently small. (See
(1.17), (1.16), (1.5) for the definitions of G, K, Γ.)
We shall now show that ψ(0) in Theorem 1.7 can be replaced by ψ0 and that ψ(t) is outgoing(in the sense described above).
Theorem 1.8 Let λ0 be an embedded eigenvalue of H0, H0ψ0 = λ0ψ0 . Assume (C0) -(C5). Then for 0 ≤ t ≤ Cκ−2 and some ǫ ∈ (0, 1] the results of Theorem 1.7 yield
for some c0 > 0, c > 0, uniformly in W ∈ Wκ0for some κ0 sufficiently small. For t ≥ 0
and some ǫ ∈ (0, 1]
‖〈A〉−αΠ0e−itHψ0‖ = O(κǫ) +O(κ) (κ→ 0) , (1.26)
uniformly in W ∈ Wκ0for some κ0 sufficiently small.
We shall prove these Theorems in Section 1.2 and Section 1.3.
Time Evolution of Quantum Resonance States 11
1.2 Proof of Theorem 1.7
In the following 〈B(H), ‖ · ‖〉 will denote the Banach space of bounded linear operators onH, σ( · ) the spectrum of an operator and C a generic positive constant, independent of κand t.
We have the following decomposition of e−iΛt:
Lemma 1.9 For 0 ≤ t ≤ Cκ−2 with some C > 0 the following is true:
e−iΛt = e−iGte−Γte−iKt + F (t) (1.27)
where B(H) ∋ F (t) = F1(t) + F2(t) + F3(t) = O(κ2t) (κ→ 0) and
F1(t) = O(κ2t) (κ→ 0) , (1.28)
F2(t) = O(κ3t) (κ→ 0) , (1.29)
F3(t) = O(κ5t2) (κ→ 0) , (1.30)
uniformly in W ∈ Wκ0for some κ0 sufficiently small.
G, K, Γ are defined in (1.17), (1.16), (1.5).
Remark 1.10 Since Γ = O(κ2) is self-adjoint on H, positive on Ran Π0 and (C5)holds, we have sup σ(Γ) = ‖Γ‖ ≤ cκ2 for some c > 0, and for φ ∈ H and t ≥ 0 we getvia functional calculus
for some c0 > 0 , c > 0. In particular for any 0 < C <∞ and 0 ≤ t ≤ Cκ−2
eΓt = O(1) (κ→ 0) , (1.32)
uniformly in W ∈ Wκ0for some κ0 sufficiently small.
To prove Lemma 1.9, we will need
Lemma 1.11 Let K = O(κ3) (κ→ 0) as in (1.14). Let ε > 0. Then for any 0 < C <∞and 0 ≤ t ≤ Cκ−3+ε we have
e±iKt = 1 +O(κε) (κ→ 0) , (1.33)
uniformly in W ∈ Wκ0for some κ0 sufficiently small.
Proof : Since K ∈ B(H) we have ‖e±iKt−1 ‖ ≤∞∑
j=1
1j! ‖Kt‖j . Then for 0 ≤ t ≤ Cκ−3+ε
we have Kt = O(κε) (κ→ 0) by (1.14). Thus
e±iKt = 1 +O(κε)∞∑
j=1
1
j!= 1 +O(κε) (κ→ 0) ,
uniformly in W ∈ Wκ0for some κ0 sufficiently small.
Proof of Lemma 1.9: In general Q, G, K and Γ do not commute. But
e−iΛt = e−iQte−iKtR(t) , (1.34)
12 J. Rama
where the operator-valued remainder R(t) := eiKteiQte−iΛt solves the initial value problemddt R(t) = i eiKt[K , eiQt] e−iΛt , R(0) = 1 . Thus
R(t) = 1 +
t∫
0
i eiKs[K , eiQs] e−iΛs ds . (1.35)
Analogously
e−iQt =: e−iGte−Γt R(t) , (1.36)
where
R(t) = 1 +
t∫
0
eΓs [Γ , eiGs] e−iQs ds . (1.37)
Using (1.34) and (1.36) we obtain
e−iΛt =(
e−iGte−Γt R(t))
e−iKtR(t) = e−iGte−Γte−iKt + F (t)
where, using (1.37) and (1.35), F (t) = F1(t) + F2(t) + F3(t) with
F1(t) := e−iGte−Γt
t∫
0
eΓs [Γ , eiGs] e−iQs ds e−iKt , (1.38)
F2(t) := e−iGte−Γte−iKt
t∫
0
i eiKs[K , eiQs] e−iΛs ds , (1.39)
F3(t) := F1(t) · eiKteΓteiGtF2(t) . (1.40)
We shall now estimate F1(t), F2(t), F3(t).
Upper bounds on F1(t): To estimate F1(t), we observe that G is self-adjoint. Next weobserve that
‖eA+iB‖ ≤ ‖eA‖ (1.41)
for A, B ∈ B(H) self-adjoint. This follows from the Lie product formula [RSim 1, TheoremVIII.29]. (We remark that the Lie product formula and its proof in [RSim 1] hold withoutany change for operators in B(H), even if dimH = ∞.) Thus, since Q ∈ B(H),
‖e−iQs‖ ≤ ‖eImQ s‖ with ImQ(1.17)= −Γ (s ≥ 0). (1.42)
By (1.38) and (1.42)
‖F1(t)‖ ≤ ‖e−Γt‖ ‖e−iKt‖t∫
0
‖eΓs‖∥∥∥[Γ , eiGs]
∥∥∥ ‖e−Γs‖ ds .
Using (1.15), we have [Γ , eiGt] = O(κ2) (κ→ 0). Using in addition Lemma 1.11 and (1.32),which holds for 0 ≤ t ≤ Cκ−2, we obtain (1.28).
Upper bounds on F2(t): Equation (1.39) combined with (1.41) yields
which proves (1.20), since G is self-adjoint. (1.21) follows from (1.31). This completes theproof of Theorem 1.7.
1.3 Proof of Theorem 1.8
A convenient functional calculus for C∞0 -functions of self-adjoint operators in Hilbert spaces
is due to B. Helffer and J. Sjostrand [HeSj], using the concept of almost analytic extensions.This calculus can be generalized to smooth functions with non-compact support, but sat-isfying certain growth conditions. Here we follow [DG, Chapter C.2 and C.3]. We use thenotations ∂ := ∂x + i∂y, C ∋ z = x+ iy.
Lemma 1.12 [DG, Proposition C.2.2]
Let ρ ∈ R. Define the following class of smooth functions:
Sρ :=f ∈ C∞(R)
∣∣ |∂k
λf(λ)| ≤ Ck〈λ〉ρ−k, k ≥ 0
(1.46)
14 J. Rama
Then for f ∈ Sρ, there exists an almost analytic extension f ∈ C∞(C) of f in thesense, that f
∣∣R
= f ,
|(∂f
)(z)| ≤ Ck〈x〉ρ−1−k|y|k , (k ∈ N) (1.47)
supp f ⊂x+ iy
∣∣ |y| ≤ C〈x〉
and
f(λ) =
∫
C
(λ− z)−1 1
2π
(∂f
)(z) dx dy . (λ ∈ R)
In particular for any self-adjoint operator T and f ∈ Sρ
f(T ) =
∫
C
(T − z)−1 1
2π
(∂f
)(z) dx dy . (1.48)
Remark 1.13 If f ∈ Sρ with compact support, we can choose f with compact sup-port, i.e. f ∈ C∞
0 (C).
We shall use the following result from [DG, Lemma C.3.2]:
Lemma 1.14 Let T , S be self-adjoint operators with∥∥[T, S]
∥∥ < ∞. If f ∈ Sρ with
ρ < 1, then
∥∥[f(T ), S ]
∥∥ ≤ C
∥∥[T, S]
∥∥
for some C <∞.
Our proof of Theorem 1.8 (respectively of Proposition 1.15 and Proposition 1.16) usesthe following expansions and estimates:
For linear operators T and S we formally have
TmS = STm +∑
j+l=mj,l≥1
cjl adjT (S)T l + adm
T (S) (1.49)
for all m ∈ N and some cjl ∈ R. Furthermore for T self-adjoint and any Borel-function f
admT ([f(T ), S]) = f(T )adm
T (S) − admT (S)f(T ) . (m ∈ N) (1.50)
The proofs of (1.49) and (1.50) are by induction.
Assume (C0) - (C3). Let g ∈ C∞0 (R), let g ∈ C∞
0 (C) be an almost analytic extension of gin the sense of Lemma 1.12. By functional calculus and (C3)
‖[H,A](H − z)−1‖ ≤ c (1 + |z|) |Im z|−1 (1.51)
for some c <∞. By induction
‖adkA
((H − z)−1
)‖ ≤ C
k+1∑
j=2
|Im z|−j (k ∈ 1, . . . , n) (1.52)
Time Evolution of Quantum Resonance States 15
for some C < ∞, uniformly for z in a bounded set. Consequently for k ∈ 1, . . . , n andsome C <∞
‖adkA
(g(H)
)‖
(1.48)
≤∫
C
‖adkA
((H − z)−1
)‖ 1
2π
∣∣(∂g
)(z)
∣∣ dx dy
(1.52)
≤ C
∫
C
k+1∑
j=2
|y|−j 1
2π
∣∣(∂g
)(z)
∣∣ dx dy
(1.47)< ∞ . (1.53)
To prove Theorem 1.8, we will need
Proposition 1.15 Let λ0 be an embedded eigenvalue of H0, H0ψ0 = λ0ψ0 . Assume(C0) - (C3). Let Ω be an interval around λ0 such that supp gΩ ⊂ ∆. Let ψ(0) :=gΩ(H)ψ0. Then ψ(0) fulfills the requirements of Theorem 1.3 and Theorem 1.7, i.e.,ψ(0) ∈ Ran (E∆(H)) ∩ D(〈A〉α) . Furthermore
ψ(0) = ψ0 +O(κ) (κ→ 0) , (1.54)
uniformly in W ∈ Wκ0for some κ0 sufficiently small.
Proof: ψ(0) ∈ Ran(E∆(H)) is obvious, since ψ(0) := gΩ(H)ψ0 and supp gΩ ⊂ ∆. To proveψ(0) ∈ D(〈A〉α), it suffices to show 〈A〉αgΩ(H)Π0 ∈ B(H).
Let N := ⌊α⌋ ∈ N be the floor of α, i.e., α = N + ε for some ε ∈ [0, 1). Let
f(A) := (A+ i)−N 〈A〉α . (1.55)
Then 〈A〉α = (A+ i)Nf(A) and f ∈ Sε ; for the definition of Sε see (1.46). Using (1.55) weget
Combining (1.58) with (1.53) and using N < n (see (C3) ), we get A1 ∈ B(H). First applying(1.49) to AN [f(A), gΩ(H)] and then using (1.50) gives
A2 = [f(A), gΩ(H)]ANΠ0 +
+∑
j+l=Nj,l≥1
clj
(
f(A)adjA(gΩ(H)) − ad
jA(gΩ(H))f(A)
)
AlΠ0 +
+f(A)adNA (gΩ(H))Π0 − adN
A (gΩ(H))f(A)Π0 . (1.60)
By (C1) and functional calculus
AkΠ0 is bounded for 0 ≤ k ≤ N ≤ α . (1.61)
Using (1.55) and (1.57) for N = 1, we obtain
‖f(A)φ‖ = ‖〈A〉εφ‖ ≤ ‖〈A〉φ‖ ≤ c(‖Aφ‖ + ‖φ‖
)(φ ∈ D(|A|) ) . (1.62)
Thus (1.62) gives
‖f(A)adjA(gΩ(H))AlΠ0‖ ≤ c
(‖Aadj
A(gΩ(H))AlΠ0‖ + ‖adjA(gΩ(H))AlΠ0‖
)
≤ c(‖adj
A(gΩ(H))Al+1Π0‖ + ‖adj+1A (gΩ(H))AlΠ0‖ + ‖adj
A(gΩ(H))AlΠ0‖)
(1.63)
and a very similar estimate for f(A)adNA (gΩ(H))Π0. Finally A2 ∈ B(H) follows from using
(1.63) in (1.60) and then taking into account (1.53), (1.58), (1.59) and (1.61).
To prove (1.54), we observe that
ψ(0) := gΩ(H)ψ0 = ψ0 +(gΩ(H) − gΩ(H0)
)Π0ψ0 , (1.64)
which follows from Π0ψ0 = ψ0 and gΩ(H0)ψ0 = ψ0. By use of (1.48) and the second resolventequation we get
(gΩ(H) − gΩ(H0)
)Π0 = −
∫
C
(H − z)−1WΠ0(H0 − z)−1 1
2π
(∂gΩ
)(z) dx dy , (1.65)
where gΩ ∈ C∞0 (C) is an almost analytic extension of gΩ in the sense of Lemma 1.12. Then
using ‖WΠ0‖ ≤ κ and (1.47) we obtain
‖(gΩ(H) − gΩ(H0)
)Π0‖
≤ κ ·∫
C
|Im z|−2 1
2π
∣∣(∂gΩ
)(z)
∣∣ dx dy = O(κ) (κ→ 0) . (1.66)
Thus (1.54) follows from (1.64) and (1.66). This completes the proof of Proposition 1.15.
We shall now show that the contribution of the dispersive part (see (1.10) ) is small, bothfor ψ(0) and ψ0. More precisely:
Proposition 1.16 Let λ0 be an embedded eigenvalue of H0, H0ψ0 = λ0ψ0 . Assume(C0) - (C5). Let Ω be an interval around λ0 such that supp gΩ ⊂ ∆. Letψ(0) := gΩ(H)ψ0, ψ(t) = e−iHtψ(0). Then:
Time Evolution of Quantum Resonance States 17
(1) For t ≥ 0 and some ǫ ∈ (0, 1] we have
‖〈A〉−αψdisp(t)‖ ≤ C ‖〈A〉αΠ0ψ(0)‖ +O(κǫ) (κ→ 0) (1.67)
for some C ≥ 0, uniformly in W ∈ Wκ0for some κ0 sufficiently small. ψdisp(t) is
defined as in (1.10). Furthermore
〈A〉αΠ0ψ(0) = O(κ) (κ→ 0) , (1.68)
uniformly in W ∈ Wκ0for some κ0 sufficiently small.
(2) For t ≥ 0
〈A〉−αψdisp,0(t) := 〈A〉−αBg∆′(H)Π0e−iHtψ0
= 〈A〉−αψdisp(t) +O(κ) (κ→ 0) , (1.69)
uniformly in W ∈ Wκ0for some κ0 sufficiently small.
Proof of Proposition 1.16, (1.67): (1.67) directly follows from (1.12).
Proof of Proposition 1.16, (1.68): From (1.64) and Π0ψ0 = 0 we get
〈A〉αΠ0ψ(0) = 〈A〉αΠ0
(gΩ(H) − gΩ(H0)
)Π0ψ0 . (1.70)
Using (1.1) we obtain
〈A〉αΠ0
(gΩ(H) − gΩ(H0)
)Π0 = 〈A〉α
(gΩ(H) − gΩ(H0)
)Π0 +O(κ) (1.71)
(κ→ 0), since by use of (C1) and (1.66)
〈A〉αΠ0
(gΩ(H) − gΩ(H0)
)Π0 = O(κ) (κ→ 0) , (1.72)
uniformly in W ∈ Wκ0for some κ0 sufficiently small. In analogy to (1.65) we get the
following estimate for the first term on the r.h.s. of (1.71):
‖〈A〉α(gΩ(H) − gΩ(H0)
)Π0‖ ≤
≤∫
C
‖〈A〉α(H − z)−1WΠ0‖ · ‖(H0 − z)−1‖ 1
2π
∣∣(∂gΩ
)(z)
∣∣ dx dy (1.73)
Thus we have to estimate ‖〈A〉α(H − z)−1WΠ0‖. Let f be defined as in (1.55). Then
We shall now prove that |Im z|νBj = O(κ) (j ∈ 1, 2, 3, 4) for some ν ∈ N , uniformlyfor z in a bounded set. Inserting (1.75) into (1.73) and using (1.47) and (1.70) - (1.72) willthen finish the proof of (1.68). By (C2) and functional calculus
‖Akf(A)WΠ0‖ ≤ κ (0 ≤ k ≤ N) , ‖AkWΠ0‖ ≤ κ (0 ≤ k ≤ α) . (1.76)
Thus we have ‖B3‖ ≤ κ|Im z|−1. Splitting (H − z)−1 into its real and imaginary parts,Lemma 1.14 together with (1.51) yields
∥∥[f(A), (H − z)−1]
∥∥ ≤ C (1 + |z|) |Im z|−2 . (1.77)
18 J. Rama
So ‖B4‖ ≤ C κ|Im z|−2 for some C ∈ R, uniformly for z in a bounded set. Applying (1.49) toAN (H − z)−1 yields
B1 = (H − z)−1ANf(A)WΠ0 +∑
j+l=Nj,l≥1
cjl adjA((H − z)−1)Alf(A)WΠ0 +
+adNA ((H − z)−1)f(A)WΠ0 .
Combining (1.52) and (1.76) leads to
‖B1‖ ≤ C κ
N+1∑
k=1
|Im z|−k
for some C < ∞, uniformly for z in a bounded set. By first applying (1.49) toAN [f(A), (H − z)−1] and then using (1.50), we obtain
B2 = [f(A), (H − z)−1]ANWΠ0 +
+∑
j+l=Nj,l≥1
cjl
(
f(A)adjA
((H − z)−1
)− ad
jA
((H − z)−1
)f(A)
)
AlWΠ0 +
+f(A)adNA
((H − z)−1
)WΠ0 − adN
A
((H − z)−1
)f(A)WΠ0 . (1.78)
Using (1.62) yields
‖f(A)adjA
((H − z)−1
)AlWΠ0‖ ≤ c
(‖adj
A
((H − z)−1
)Al+1WΠ0‖ +
+‖adj+1A
((H − z)−1
)AlWΠ0‖ + ‖adj
A
((H − z)−1
)AlWΠ0‖
)(1.79)
and a similar estimate for f(A)adNA
((H − z)−1
)WΠ0. Thus, combining (1.52), (1.76) and
(1.77) with (1.78) and (1.79) and using N + 1 < n (see (C3)), we obtain
‖B2‖ ≤ C1 κ
N+1∑
k=1
‖adkA
((H − z)
)−1‖ ≤ C2 κ
N+2∑
k=2
|Im z|−k
for some C1 <∞, C2 <∞, uniformly for z in a bounded set.
Proof of Proposition 1.16, (1.69): Since Bg∆′(H)Π0e−itH ∈ B(H) (cp. (1.1), (1.2), (1.9)),
substitution of (1.54) into (1.10) yields
ψdisp(t) = Bg∆′(H)Π0e−itHψ0 +O(κ) (κ→ 0) , (1.80)
uniformly in W ∈ Wκ0for some κ0 sufficiently small. Thus (1.69) follows from (1.80). This
completes the proof of the proposition.
Now we are prepared to give the
Proof of Theorem 1.8: (1.22) – (1.24) follow from using (1.54) in (1.18) – (1.20). Theestimate (1.25) directly follows from (1.31). For the proof of (1.26) we need Proposition1.16: By use of (1.9) and (1.10) we obtain
ψdisp(t) = g∆′(H)Π0ψ(t) +O(κ)
(1.3)= Π0ψ(t) − g∆′(H)Π0ψ(t) +O(κ) (κ→ 0) ,
uniformly in W ∈ Wκ0for some κ0 sufficiently small. Since ψ(0) ∈ RanE∆(H), we have
for t ≥ 0, uniformly in W ∈ Wκ0for some κ0 sufficiently small. [MeSi, Proposition 3.1] gives
g∆′(H)Π0g∆(H) = O(κ) (κ→ 0). Thus for t ≥ 0
ψdisp(t) = Π0ψ(t) +O(κ) (κ→ 0) , (1.81)
uniformly in W ∈ Wκ0for some κ0 sufficiently small. Combining (1.68) and (1.67), we get
for t ≥ 0 and some ǫ ∈ (0, 1]
‖〈A〉−αψdisp(t)‖ = O(κǫ) +O(κ) (κ→ 0) , (1.82)
uniformly in W ∈ Wκ0for some κ0 sufficiently small. Substitution of (1.81) into (1.82) gives
‖〈A〉−αΠ0ψ(t)‖ = O(κǫ) +O(κ) (κ→ 0) (1.83)
for t ≥ 0, uniformly in W ∈ Wκ0for some κ0 sufficiently small. Finally inserting (1.54) into
(1.83) yields (1.26). This completes the proof of Theorem 1.8.
Part 2
Time Evolution of Quantum Resonance States II:A Stationary Approach Based on Analyticity
2.1 Introduction
In this Part 2 we sharpen the results of Part 1: We prove – under stronger conditions on theHamiltonian – that the perturbed dynamics of an embedded bound state (corresponding toa degenerate embedded eigenvalue of the unperturbed Hamiltonian) are dominated by anexponential decay for times exceeding the expected lifetime. Although our results are basedon the dilation analytic theory of [Hu2], our methods and results seem also to be applicableto non dilation analytic cases; see Part 3 for a discussion.
Let H(κ) (κ ≥ 0 small) be a family of self-adjoint operators in a complex Hilbert space,satisfying an abstract dilation analytic setting (or: abstract Balslev-Combes setting): ForU(θ) (θ ∈ R) some strongly continuous one parameter unitary group the relation H(κ, θ) :=U(θ)H(κ)U(θ)−1 (θ ∈ R), which holds for fixed κ ≥ 0 small, extends analytically for θin some complex strip; see (A1). Furthermore, for some non-real fixed θ let H(κ, θ) beanalytic (in the generalized sense) in the variable κ for κ in some complex neighborhoodof 0; see (A2). Let λ0 be a finitely degenerate eigenvalue of H0 := H(κ)|κ=0 embeddedin the essential spectrum σess(H0) with corresponding eigenprojection Π0. For non-realdilation parameter θ the embedded eigenvalue λ0 of H0 becomes a discrete eigenvalue ofthe dilated unperturbed Hamiltonian H0(θ), and the resonance eigenvalues are defined tobe the complex eigenvalues of the dilated perturbed (not self-adjoint) Hamiltonian H(κ, θ)splitting from the discrete eigenvalue λ0 of H0(θ).
For dilation analytic perturbations V and H(κ) = H0 + κV (see (2.19)) we analyze theasymptotics Π0e
−itH(κ)Π0 in the limit κ → 0 for the special case dimRanΠ0 = 2. Some ofour results also persist in case of arbitrary dim RanΠ0 <∞. Independent of dimRanΠ0 <∞we get the operator relation on RanΠ0
(see (2.172)), where h(κ) is some (not self-adjoint) analytic matrix family. For dimRanΠ0 =2 it turns out that ‖Π0e
−itH(κ)Π0‖ decays – up to a time-dependent remainder – roughly
exponentially driven by a term e−κ2γt (see Theorem 2.41, 2.), where γ is some strictly positiveconstant determined by the imaginary parts of the resonance eigenvalues. By constructionthe resonance eigenvalues coincide with the eigenvalues of h(κ). The key tool to obtain theseresults is a Jordan decomposition of the analytic matrix family h(κ).
For times 0 ≤ t = O((− lnκ)κ−2) (κ → 0) the time-dependent remainder is dominatedby the exponential term; see Corollary 2.42. This is a logarithmic improvement of timecontrol with regard to the results of Theorem 1.8. This may seem weak, but really needs alot of analytic structure and is in fact the central result. Furthermore, for fixed κ > 0 smallenough it holds limt→∞ ‖Π0e
−itH(κ)Π0‖ = O(κ2); see (2.177).
21
22 J. Rama
Under the assumptions (A1) - (A6) outlined in Section 2.1.1 our results follow by extendingHunziker’s results in [Hu2]. Under conditions almost equivalent to ours the paper [Hu2]treats resonance states which appear as perturbed bound states in the abstract Balslev-Combes theory sketched above. (In fact the paper [Hu2] does not need our assumption (A2);so our results hold for a subclass of Hamiltonians considered in [Hu2].) The correspondingmetastable states are constructed using a (formal) Rayleigh-Schrodinger expansion to orderN − 1 for the (nonexistent) perturbed bound state. This setting applies in arbitrary orderN ≥ 1 to cases like the stark effect and in lowest order N = 1 to perturbation of boundstates embedded in the continuum. It is the latter case we have taken up. (And we remarkthat our assumption (A2) makes all occurring Rayleigh-Schrodinger expansion well-definedand convergent): Assume our conditions (A1) - (A6). In particular let λ0 be an embeddedeigenvalue of H0 with corresponding eigenprojection Π0, H0ψ0 = λ0ψ0, ψ0 normalized.
If λ0 is simple, applied to our setting the results of [Hu2] give explicit asymptotics for theeigenfunction ψ0 under the perturbed dynamics e−itH(κ): In the sense of (2.75) the ”naive”resonance state e−itH(κ)ψ0 shows an exponential decay up to a background term of orderO(κ2) as κ→ 0.
Things become more complicated, if λ0 is degenerate: If λ0 is degenerate, the decay lawfor states in RanΠ0 under the perturbed dynamics e−itH(κ) is set by the operator-valuedfunction e−ith(κ) up to a time-dependent remainder of order O(κ2) (κ → 0) in the sense ofCorollary 2.11. Explicit asymptotics for such states have not been worked in [Hu2].
We shall now precisely formulate the assumptions under which our results of Section 2.3hold.
2.1.1 Assumptions
For the sake of the reader in the following we amplify the presentation in [Hu2].
(A1) Abstract Balslev-Combes Setting:
For real κ ≥ 0 small enough let H(κ) be a family of self-adjoint operators in a complexHilbert space 〈H, ‖ · ‖〉. Let U(θ) (θ ∈ R) be a strongly continuous one parameterunitary group, such that for fixed κ ≥ 0 the relation
H(κ, θ) := U(θ)H(κ)U(θ)−1 (θ ∈ R) (2.1)
extends analytically into the strip
Sβ :=θ ∈ C
∣∣ |Imθ| < β
for some β > 0, where
H(κ, θ)∗ = H(κ, θ) (θ ∈ Sβ)
holds. The spectrum of H(κ, θ) depends only on Imθ and is assumed to lie in theclosed lower halfplane for Imθ > 0.
For θ ∈ Sβ the group U(θ) is defined by spectral calculus, having the (formal) repre-sentation
U(θ) :=
∫
R
e−iµθ dEµ (θ ∈ Sβ) (2.2)
and satisfying
U(θ)∗ = U(θ)−1 (θ ∈ Sβ) (2.3)
on some natural dense domain (depending on θ) in H.
Time Evolution of Quantum Resonance States 23
We remark that the analytic continuation of (2.1) into the strip Sβ is given by
H(κ, θ) = U(θ)H(κ)U(θ)−1 (θ ∈ Sβ) . (2.4)
We add the following assumption (A2), which is not an assumption in [Hu2]:
(A2) For some fixed θ ∈ Sβ with Imθ > 0 the family H(κ, θ) is analytic (in the generalizedsense) in the variable κ for κ in some complex neighborhood of zero.
(For the definition of ”analytic in the generalized sense” see Definition F.2.)
We abbreviate
H0 := H(κ, θ)∣∣κ=0, θ=0
, H0(θ) := H(κ, θ)∣∣κ=0
, R0(θ, z) := (H0(θ) − z)−1 .
Note that by (A1) the operator H0 is self-adjoint in H.
(A3) Let λ0 be an eigenvalue of H0 embedded in its essential spectrum σess(H0) withcorresponding eigenprojection Π0, dimRanΠ0 =: N <∞. Assume
λ0 ∈ σdisc(H0(θ)) (θ ∈ Sβ\R) ,
where σdisc(H0(θ)) denotes the discrete spectrum of H0(θ).
Consequences of (A3): As a consequence of the analyticity of H0(θ) the correspondingeigenprojection
Π0(θ) := − 1
2πi
∮
γθ
R0(θ, z) dz(θ ∈ θ′ ∈ Sβ | Imθ′ 6= 0
), (2.5)
with γθ some curve in the resolvent set ρ(H0(θ)), enclosing this λ0 ∈ σdisc(H0(θ)) but noother points of the spectrum σ(H0(θ)), is analytic in the full strip Sβ. In particular one gets
dim RanΠ0(θ) = dim RanΠ0 = N <∞ (θ ∈ Sβ) (2.6)
by use of [K, I § 4.6, Lemma 4.10]. A further consequence of (A3) is that
Π0(θ) = U(θ)Π0U(θ)−1 (θ ∈ Sβ) (2.7)
holds on the dense domain of U(θ)−1. Thus RanΠ0 is contained in the domain of U(θ)(θ ∈ Sβ) and
U(θ) RanΠ0 : RanΠ0 → RanΠ0(θ) , (2.8)
U(θ)−1 RanΠ0(θ) : RanΠ0(θ) → RanΠ0 (2.9)
are bounded operators for θ ∈ Sβ.
Now we fix some θ ∈ Sβ with Imθ 6= 0 and consider the perturbation of λ0 ∈ σdisc(H0(θ))by H(κ, θ) −H0(θ). We require stability of λ0 under perturbation in the following sense:
(A4) Stability conditions on λ0:
(i) For fixed θ ∈ Sβ\R there exists a punctured neighborhood
W (λ0; θ)• := z ∈ C | 0 < |z − λ0| < rθ for some rθ > 0 (2.10)
of λ0 such that
R(κ, θ, z) :=(H(κ, θ) − z
)−1(z ∈W (λ0; θ)
•, 0 ≤ κ < κ0(z)) (2.11)
24 J. Rama
exists and is uniformly bounded for each fixed z ∈W (λ0; θ)• and 0 ≤ κ < κ0(z),
where
κ0(·) : W (λ0; θ)• → R
+
z 7→ κ0(z)
is some continuous map. (Note that R(κ, θ, z) is continuous in the variable κ, ifassumption (A2) is fulfilled. Thus (A2) proves the existence of some continuousκ0(·).)
(ii) Further assume that for fixed θ ∈ Sβ\R the perturbed total Riesz projection (forall eigenvalues of H(κ, θ) splitting from λ0, i.e., the λ0-group of H(κ, θ) in thesense of Kato [K, p.66])
Π(κ, θ) := − 1
2πi
∮
Γθ
R(κ, θ, z) dz (0 ≤ κ < κ0(z)) (2.12)
with Γθ an arbitrary loop around λ0 in W (λ0; θ)• satisfies
limκ→0
‖Π(κ, θ) − Π0(θ)‖ = 0 . (2.13)
Consequences of (A4): For fixed θ ∈ Sβ\R the resolvent R(κ, θ, z) is uniformly boundedfor z ∈ Γθ and 0 ≤ κ < κ0(Γθ). This is a consequence of (i). Equation (2.13) implies
dimRanΠ(κ, θ) = dimRanΠ0(θ)(2.6)= dim RanΠ0 (2.14)
for κ small enough. (The property (2.14) can be seen by [K, I § 4.6, Lemma 4.10].) Thereforeλ0 is the limit as κ → 0 of a group of perturbed eigenvalues λ(κ) having total algebraicmultiplicity N = dimRanΠ0. These are the eigenvalues of the reduced operator
Remark 2.1 By convention we denote with λ(κ) the eigenvalues of H(κ, θ) for Imθ >0. These are the resonances corresponding to the unperturbed eigenvalue λ0 = λ(0).As we will see later, these resonance eigenvalues splitting from λ0 can be groupedinto cycles. We will then write λρ,l,i(κ) for the eigenvalue forming the i-th branchof the l-th cycle (corresponding to an irreducible polynomial χρ in a factorizationof the characteristic polynomial of h(κ); confer with (2.81)). All this is derived inAppendix C.
Remark 2.2 The assumption (A1) on the spectrum σ(H(κ, θ)) for θ ∈ Sβ with Imθ >0 implies
Imλ(κ) ≤ 0 . (2.16)
This is proven as follows: By convention λ(κ) denotes an eigenvalue of H(κ, θ)(2.15)=
Π(κ, θ)H(κ, θ)Π(κ, θ) for Imθ > 0. By assumption (A1) it holds σ(H(κ, θ)) ⊂ C− :=
Remark 2.3 The spectrum of H(κ, θ) depends only on Imθ in the following sense: Letκ ≥ 0 be small enough and fixed. Assume a resonance eigenvalue λ(κ) of H(κ, θ1) withsome Imθ1 > 0. If Imθ1 varies just a little bit, say Imθ1 changes into some Imθ2 > 0with |Imθ1 − Imθ2| small enough, then the resonance eigenvalue λ(κ) persists, i.e.,λ(κ) ∈ σ(H(κ, θ2)).
As remarked in [Hu2] on p.180 before equation (9), the spectrum σ(H(κ, θ)) dependsonly on the sign of Imθ. But in fact, we see no reason, why a big change of Imθ(even if the sign of Imθ is preserved) should not change σ(H(κ, θ)): Further resonanceeigenvalues might occur if |Imθ| is growing, respectively resonance eigenvalues mightbe ”covered” if |Imθ| is decreasing. The relations
Π(κ, θ2) = U(θ2 − θ1)Π(κ, θ1)U(θ2 − θ1)−1 ,
H(κ, θ2) = U(θ2 − θ1)H(κ, θ1)U(θ2 − θ1)−1
hold for κ small, if Imθ1 and Imθ2 have the same sign; see [Hu2, (9)].
We restrict the self-adjoint family H(κ) considered in (A1) to the following form:
(A5) For θ ∈ Sβ and κ ≥ 0 small enough let
κ · V (θ) := H(κ, θ) −H0(θ)
be a densely defined closed operator in H with
V (θ)∗ = V (θ) (θ ∈ Sβ) , (2.18)
such that
H(κ, θ) = H0(θ) + κV (θ) (θ ∈ Sβ) (2.19)
holds on a core of H(κ, θ).
Remark 2.4 The operator V := V (θ)|θ=0 is symmetric in H by (2.18). One obtains
V (θ) = U(θ)V U(θ)−1 (θ ∈ Sβ) , (2.20)
H0(θ) = U(θ)H0U(θ)−1 (θ ∈ Sβ) (2.21)
by inserting (2.19) for θ ∈ R into (2.1) and then taking into account analytic continu-ation into Sβ . Note that (2.21) also directly follows from (A1) with κ = 0.
(A6) For fixed θ ∈ Sβ the operator V (θ) is relatively bounded with respect to H0(θ).V (θ)Π0(θ) is analytic in θ ∈ Sβ.
Consequences of (A6): The second resolvent equation
is valid for θ ∈ Sβ with Imθ 6= 0, z ∈ W (λ0; θ)• and small κ ≥ 0. V (θ)Π0(θ) ∈ B(H)
(θ ∈ Sβ).
Remark 2.5 To discuss resonances which appear as perturbed bound states Hunziker[Hu2] uses dilation analyticity and perturbation theory in the spirit of [Sim3], wherethe group U(θ) is the dilation group (see (F.2) and (F.3)) and the perturbation V
is assumed to be in the class Cβ (see Definition F.7). In particular V (θ) is boundedwith respect to H0(θ). Although the abstract Balslev-Combes setting stated in (A1)
26 J. Rama
does not restrict the potentials V to the class Cβ , the Rayleigh-Schrodinger expansionsin [Hu2] (stated here in Section 2.1.2) explicitly need perturbations V (θ) relativelybounded with respect to H0(θ) in the sense of (A6).
We expect the qualitative statements of the results in [Hu2] (and therefore the resultsin this thesis) to persist in the case of H0-form-compact perturbations V ∈ Fβ (seeDefinition F.8). This needs new expansions of Π(κ, θ) similar to those of (2.22) butadapted to the setting of Fβ . For example, if V (θ) is just form-bounded or form-compact with respect to H0(θ) the expression H(κ, θ)Π0(θ) (appearing in, e.g., (2.53))would in general not even be well-defined.
2.1.2 Stating Hunziker’s Results
Assume (A1) - (A6).
As a consequence of (A4) (i) for fixed θ ∈ Sβ\R the resolvents R(κ, θ, z) and R0(θ, z) areuniformly bounded for z ∈ Γθ and 0 ≤ κ < z0(Γθ). Thus for fixed θ ∈ Sβ\R their difference,given by the second resolvent equation
is also uniformly bounded for z ∈ Γθ and 0 ≤ κ < κ0(Γθ). Then by use of (2.23) in (2.12)one obtains
Π(κ, θ) = − 1
2πi
∮
Γθ
R0(θ, z) dz +1
2πi
∮
Γθ
κR(κ, θ, z)V (θ)R0(θ, z) dz
= Π0(θ) +1
2πi
∮
Γθ
κR(κ, θ, z)V (θ)R0(θ, z) dz (2.24)
for fixed θ ∈ Sβ\R and all 0 ≤ κ < κ0(Γθ). The integral on the r.h.s. in (2.24) is an O(κ) asκ→ 0. This can be seen as follows: As a consequence of (A4) (i) the resolvent R(κ, θ, z) isuniformly bounded for z ∈ Γθ and 0 ≤ κ < κ0(Γθ), and V (θ)R0(θ, z) is bounded for z ∈ Γθ
as a consequence of (A6). Thus
‖R(κ, θ, z)V (θ)R0(θ, z)‖ ≤ C (z ∈ Γθ , 0 ≤ κ < κ0(Γθ))
for some 0 ≤ C <∞ and therefore
∥∥κ
2πi
∮
Γθ
R(κ, θ, z)V (θ)R0(θ, z) dz∥∥ ≤ κ
2π|Γθ| max
z∈Γθ
‖R(κ, θ, z)V (θ)R0(θ, z)‖
= O(κ) (κ→ 0) . (2.25)
Thus by use of (2.25) in (2.24) we get
Π(κ, θ) = Π0(θ) +O(κ) (κ→ 0) (2.26)
for fixed θ ∈ Sβ\R and all 0 ≤ κ < z0(Γθ).
Remark 2.6 In stationary perturbation theory a Rayleigh-Schrodinger expansion isa (possibly just formal) power series expansion of eigenfunctions or eigenvalues inthe perturbation parameter. The results in [Hu2] go back to a Rayleigh-Schrodingerexpansion of Π(κ, θ) in the parameter κ.
Let θ ∈ Sβ\R be fixed. From (A2) and Theorem F.4 follows that for all z ∈ ρ(H0(θ))there exists rθ(z) > 0 such that the resolvent R(κ, θ, z) = (H(κ, θ)− z)−1 is bounded-analytic in the two variables z and κ on the set
〈κ, z〉
∣∣ z ∈ ρ(H0(θ)) , |κ| < rθ(z)
.
Time Evolution of Quantum Resonance States 27
Thus Π(κ, θ), given by (2.12), is bounded-analytic in the complex variable κ for |κ|sufficiently small. This holds, since
∂
∂κ
∮
Γθ
R(κ, θ, z) dz =
∮
Γθ
∂
∂κR(κ, θ, z) dz .
(The proof uses dominate convergence and the definition of the derivative.) ThusΠ(κ, θ) has a convergent Rayleigh-Schrodinger expansion, i.e., a convergent (and uniquelydetermined) power series expansion in the variable κ for |κ| small enough.
Remark 2.7 In Hunziker’s notation (slightly modified; see [Hu2, p.181])
Π(κ, θ) = P (N)(κ, θ) +O(κN ) (κ→ 0) (2.27)
for κ ≥ 0 small enough and θ ∈ Sβ\R fixed. P (N)(κ, θ) denotes the perturbativeexpression up to order N−1 in the Rayleigh-Schrodinger expansion of Π(κ, θ). So thisN in [Hu2] has nothing to do with our N = dim RanΠ0.
According to (2.27) with N = 1 and Remark 2.6 it holds
where P (1)(κ, θ) is a power series expansion in the variable κ of Π(κ, θ) up to order 1−1 = 0.So P (1)(κ, θ) is the 0-th coefficient in this expansion and therefore independent of κ anduniquely determined. Then comparing equations (2.26) and (2.28) yields
P (1)(κ, θ) = Π0(θ) (2.29)
for fixed θ ∈ Sβ\R and all 0 ≤ κ < κ0(Γθ). In particular P (1)(κ, θ) is analytic in θ ∈ Sβ
(since Π0(θ) is analytic in θ ∈ Sβ, which is a consequence of (A3)) and
P (1)(κ) := P (1)(κ, θ)|θ=0 = Π0 (0 ≤ κ < κ0(Γθ)) .
has been defined in [Hu2, (17)] for θ ∈ Sβ fixed with Imθ > 0. Using (2.29) in (2.30) gives
D(1)(κ) = U(θ)−1 Π0(θ)Π(κ, θ)Π0(θ)︸ ︷︷ ︸
(⋆)
U(θ) (2.31)
for θ ∈ Sβ fixed with Imθ > 0. Obviously, D(1)(κ) is an endomorphism on RanΠ0. Asremarked in [Hu2] the r.h.s. in (2.30) depends only on the sign of Imθ. For Imθ < 0definition (2.30) remains valid if D(1)(κ) is replaced by its adjoint D(1)(κ)∗. The expression(⋆) in (2.31) is an approximate Π(κ, θ). A simple calculation shows the identity
Π0(θ)Π(κ, θ)Π0(θ) = Π0(θ) −(Π0(θ) − Π(κ, θ)
)(1 − Π(κ, θ)
)(Π0(θ) − Π(κ, θ)
)(2.32)
for this expression. Using Π0(θ) − Π(κ, θ)(2.26)= O(κ) (κ→ 0) in (2.32) implies
Π0(θ)Π(κ, θ)Π0(θ) = Π0(θ) +O(κ2) (κ→ 0) (2.33)
for fixed θ ∈ Sβ with Imθ > 0 and all 0 ≤ κ < κ0(Γθ). Then inserting (2.33) into (2.31)leads to
which we will need later on. Equation (2.35) can be shown by a binomial series expansion
(1 +A(κ))±1/2 =∞∑
ν=0
(±12
ν
)
A(κ)ν = 1 +O(κ2) (κ→ 0) (2.36)
on RanΠ0, if A(κ) = O(κ2). This series is absolutely convergent, since ‖A(κ)‖ < 1 for κsmall enough. In the notation of Hunziker (slightly modified; cf. [Hu2, (18)]), it is
D(1)(κ) = d(1)(κ) +O(κ2) (κ→ 0) , (2.37)
where the symbol d(1)(κ) denotes the power series expansion of D(1)(κ) up to order 1−1 = 0,which goes back to the Rayleigh-Schrodinger expansion of Π(κ, θ); cf. (2.30) respectively(2.31). So d(1)(κ) is uniquely determined and independent of κ. Using this together with(2.34) proves
d(1)(κ) = Π0 . (2.38)
For θ ∈ Sβ fixed with Imθ > 0 and κ small the operator
T (κ, θ) := Π(κ, θ)P (1)(κ, θ)Π0(θ)U(θ)(D(1)(κ)
)−1/2: RanΠ0 → RanΠ(κ, θ) , (2.39)
given in [Hu2, (19)], has the inverse
T (κ, θ)−1 =(D(1)(κ)
)−1/2U(θ)−1Π0(θ)P
(1)(κ, θ)Π(κ, θ) : RanΠ(κ, θ) → RanΠ0 ; (2.40)
see [Hu2, (20)]. T (κ, θ)±1 are used to transform H(κ, θ) into the equivalent operator
which is independent of θ for Imθ > 0; see [Hu2, (21)]. By construction, the eigenvalues ofh(κ) are the resonances λ(κ); see Remark 2.1. We will now prove
for θ ∈ Sβ fixed with Imθ > 0 and all 0 ≤ κ < κ0(Γθ). Inserting (2.43) and (2.44) into (2.41)leads to
h(κ) =(D(1)(κ)
)−1/2U(θ)−1Π0(θ)H(κ, θ)Π0(θ)U(θ)
(D(1)(κ)
)−1/2(2.45)
= A(κ, θ) +B(κ, θ) , (2.46)
where
A(κ, θ) :=(D(1)(κ)
)−1/2U(θ)−1Π0(θ)H(κ, θ)Π0(θ)U(θ)
(D(1)(κ)
)−1/2, (2.47)
B(κ, θ) :=(D(1)(κ)
)−1/2U(θ)−1Π0(θ)
(1 − Π(κ, θ)
)
H(κ, θ)(1 − Π(κ, θ)
)Π0(θ)U(θ)
(D(1)(κ)
)−1/2(2.48)
Time Evolution of Quantum Resonance States 29
for θ ∈ Sβ fixed with Imθ > 0 and all 0 ≤ κ < κ0(Γθ). To get the decomposition of h(κ)described by (2.46) - (2.48), we used the identity
H(κ, θ)(2.17)= H(κ, θ) −
(1 − Π(κ, θ)
)H(κ, θ)
(1 − Π(κ, θ)
). (2.49)
Next we estimate A(κ, θ) and B(κ, θ): Inserting (2.4), (2.7) and (2.35) into the definition(2.47) one obtains
A(κ, θ) =(Π0 +O(κ2)
)U(θ)−1U(θ)Π0U(θ)−1U(θ)H(κ)U(θ)−1
U(θ)Π0U(θ)−1U(θ)(Π0 +O(κ2)
)
= Π0H(κ)Π0 +O(κ2) (κ→ 0) , (2.50)
which is an operator relation on RanΠ0 independent of θ for Imθ > 0. By use of (2.35) in(2.48) one gets
B(κ, θ) =(Π0 +O(κ2)
)U(θ)−1Π0(θ)
(1 − Π(κ, θ)
)
H(κ, θ)(1 − Π(κ, θ)
)Π0(θ)U(θ)
(Π0 +O(κ2)
)(2.51)
on RanΠ0 for θ ∈ Sβ fixed with Imθ > 0 and all 0 ≤ κ < κ0(Γθ). Note that by use of (2.4),(2.7) and (2.26) the r.h.s. in (2.51) is again independent of θ for Imθ > 0. It holds
Π0(θ)(1 − Π(κ, θ)
) (2.26)= Π0(θ)
(1 − Π0(θ) +O(κ)
)= O(κ) (κ→ 0) (2.52)
for θ ∈ Sβ fixed with Imθ > 0 and all 0 ≤ κ < κ0(Γθ). Further we get
H(κ, θ)(1 − Π(κ, θ)
)Π0(θ) =
(1 − Π(κ, θ)
)H(κ, θ)(Π0(θ) − Π(κ, θ))Π0(θ) (2.53)
for θ ∈ Sβ fixed with Imθ > 0 and all 0 ≤ κ < κ0(Γθ). Next we will show
for θ ∈ Sβ fixed with Imθ > 0 and all 0 ≤ κ < κ0(Γθ): By use of (2.5), (2.12) and the secondresolvent equation (2.23) one gets
H(κ, θ)(Π0(θ) − Π(κ, θ))Π0(θ) = H(κ, θ)1
2πi
∮
Γθ
(R(κ, θ, z) −R0(θ, z)) dzΠ0(θ)
= − κ
2πi
∮
Γθ
(H(κ, θ) − z + z)R(κ, θ, z)V (θ)R0(θ, z) dzΠ0(θ)
= − κ
2πi
∮
Γθ
V (θ)R0(θ, z) dzΠ0(θ) −κ
2πi
∮
Γθ
z R(κ, θ, z)V (θ)R0(θ, z) dzΠ0(θ)
= κV (θ)Π0(θ) −κ
2πi
∮
Γθ
z R(κ, θ, z)V (θ)R0(θ, z) dzΠ0(θ) (2.55)
for θ ∈ Sβ fixed with Imθ > 0 and all 0 ≤ κ < κ0(Γθ). As a consequence of (A6) theoperator V (θ)Π0(θ) is bounded for θ ∈ Sβ fixed with Imθ > 0. And as a consequence of(A4) (i) the resolvent R(κ, θ, z) is uniformly bounded for for z ∈ Γθ and 0 ≤ κ < κ0(Γθ).Thus for fixed θ ∈ Sβ with Imθ > 0
‖H(κ, θ)(Π0(θ) − Π(κ, θ))Π0(θ)‖(2.55)
≤ κ ‖V (θ)Π0(θ)‖+
κ
2π|Γθ| max
z∈Γθ
‖z R(κ, θ, z)V (θ)R0(θ, z)‖
≤ C κ (2.56)
30 J. Rama
for some 0 ≤ C <∞ and all 0 ≤ κ < κ0(Γθ). This proves (2.54). Then using Π(κ, θ) ∈ B(H)and (2.54) in (2.53) proves
H(κ, θ)(1 − Π(κ, θ)
)Π0(θ) = O(κ) (κ→ 0) (2.57)
for fixed θ ∈ Sβ with Imθ > 0 and 0 ≤ κ < κ0(Γθ). Then inserting (2.52) and (2.57) into(2.51) implies
B(κ, θ) = O(κ2) (κ→ 0) (2.58)
for fixed θ ∈ Sβ with Imθ > 0 and all 0 ≤ κ < κ0(Γθ) as an operator on RanΠ0, which isthen independent of θ for Imθ > 0. Finally inserting (2.50) and (2.58) into (2.46) proves(2.42).
Obviously, for (real) κ ≥ 0 small enough Π0H(κ)Π0 is a family of bounded self-adjointoperators on RanΠ0, but note that h(κ) is in general not self-adjoint: From (2.42) follows
h(κ)∗ − h(κ) = O(κ2) (κ→ 0) (2.59)
for κ ≥ 0 small enough; see also [Hu2, (24)]. This gives the estimate
0(2.16)
≥ Imλ(κ) = O(κ2) (κ→ 0) (2.60)
for κ ≥ 0 small; see [Hu2, (25)].
Next we cite Hunziker’s Theorem [Hu2, Theorem 2] for the formal perturbation expansionin lowest order (i.e., N = 1 in the notation of [Hu2]):
Theorem 2.8 (cf. [Hu2, Theorem 2])
Assume (A1) and (A3) - (A6). Let ∆ be an interval containing λ0 but no othereigenvalue of H0 different from λ0. Let g∆ ∈ C∞
0 be a smoothed out version of thecharacteristic function 1∆, i.e.
g∆(µ) =
1 , µ ∈ ∆0 , µ outside some neighborhood of ∆
,
such that supp g∆ also contains no other eigenvalue of H0 different from λ0. Then thefollowing operator relation holds on RanΠ0:
for all n ≥ 0 and some corresponding constant cn.(The operators D(1)(κ) and h(κ), which are endomorphisms on RanΠ0, are defined in(2.30) and (2.41). P (1)(κ) has been introduced in (2.28) respectively in Remark 2.7.)
Remark 2.9 Assume (A1) - (A6). By inserting (2.29) and (2.35) into (2.61) the resultof Theorem 2.8 is equivalent to the following operator relation on RanΠ0:
where λ(κ) denotes the resonance eigenvalue (which is in particular an eigenvalue of h(κ))”splitting” from λ0; cf. Remark 2.1 and the note below (2.41).
Assume (A1) - (A6). If λ0 is degenerate (and thus dimRanΠ0 > 1), the decay law ofCorollary 2.11 does not give explicit asymptotics for states in RanΠ0 under the perturbeddynamics e−itH(κ).
We will develop such explicit asymptotics in case of a degenerate λ0, dim RanΠ0 = 2, in thenext sections using a Jordan decomposition of the matrix family h(κ).
2.2 The Dynamics e−ith(κ)
Under the assumptions (A1) - (A6) our goal is to derive a direct dynamical interpretationin Hilbert space for the operator relation (2.63), which is in essential the dynamical inter-pretation of e−ith(κ). We will give this dynamical interpretation in Proposition 2.40 and inour main Theorem 2.41. Crucial for the proof of Proposition 2.40 and Theorem 2.41 is theanalyticity of h(κ):
2.2.1 Analyticity of h(κ)
Theorem 2.12 Assume (A1) - (A6). Then the operators h(κ) ∈ End(RanΠ0) andD(1)(κ) ∈ End(RanΠ0) found in Theorem 2.8 are bounded-analytic for κ in somecomplex neighborhood of zero.
(See Definition F.1 for ”bounded-analytic”.)
This result has not been formulated in [Hu2], perhaps since in [Hu2] the main emphasis hasnot been put on analytic perturbations of embedded eigenvalues.
Proof of Theorem 2.12: As shown in Remark 2.6 the projection Π(κ, θ), given by (2.12), isbounded-analytic in the complex variable κ for |κ| sufficiently small. Lemma F.3 yields thatthis is equivalent to Π(κ, θ) being analytic in the generalized sense in the variable κ for |κ|small enough.
H(κ, θ) is analytic in the generalized sense by assumption (A2). Thus H(κ, θ) defined by(2.15) is analytic in the generalized sense, since Π(κ, θ) and H(κ, θ) are analytic in thegeneralized sense. Obviously, H(κ, θ) is a bounded operator, and so by use of Lemma F.3H(κ, θ) is bounded-analytic for |κ| small enough.
D(1)(κ) is bounded-analytic for |κ| sufficiently small, since (2.31) holds and Π(κ, θ) isbounded-analytic for |κ| sufficiently small.
T (κ, θ) and T (κ, θ)−1, given by (2.43) and (2.44), are bounded-analytic for |κ| sufficientlysmall, since Π(κ, θ) and D(1)(κ) are bounded-analytic for |κ| sufficiently small.
Finally, h(κ)(2.41):= T (κ, θ)H(κ, θ)T (κ, θ)−1 is bounded-analytic for |κ| sufficiently small,
since T (κ, θ)±1 and H(κ, θ) are bounded-analytic for |κ| sufficiently small.
Time Evolution of Quantum Resonance States 33
2.2.2 Jordan Decomposition and Characteristic Polynomial of theAnalytic Matrix Family h(κ)
We are now going to analyze the analytic family h(κ) in more detail. Setting κ = 0 in (2.42)yields
h0 := h(0) = Π0H0Π0(A3)= λ0Π0 . (2.76)
Thus in particular we have
h0 = λ01 RanΠ0 . (2.77)
According to Theorem 2.12 the family h(κ) is bounded-analytic in some open connectedsubset of C containing κ = 0. Let X ⊂ C, 0 ∈ X, denote this open connected subset. So
h(·) : X → End(Ran Π0)
κ 7→ h(κ) :=
∞∑
ν=0
κνh(ν) , h(0) = h(0) =: h0 , (2.78)
where h(ν) ∈ End(RanΠ0) denotes the ν-th Taylor coefficient of h(κ). The characteristicequation det(h(κ) − λ) = 0 is an algebraic equation in λ of degree N = dim Ran Π0 withcoefficients being analytic functions on X; see [K, II § 1.1]. (We refer the reader to AppendixC, where we have collected all results on algebraic equations important in our context.) Wehave
(For the definitions of O(X)[λ] and M(X)[λ] see Appendix C.1.) We will analyze solutionsof (2.79) for κ near κ = 0 (which are the eigenvalues of h(κ) splitting from λ0) and thereforeuse the following factorization theorem:
Theorem 2.13 [Bau, part of Anhang § 2.6 Theorem 1]
Let X ⊂ C be an open connected subset.
1. Let P (λ) ∈ M(X)[λ] be normalized. Then there exists a unique factorization
P (λ) =
r∏
ρ=1
Pρ(λ)mρ (2.80)
for some r ≤ degree(P (λ)) and some mρ ∈ N, Pρ(λ) ∈ M(X)[λ] normalized and irre-ducible, Pρ(λ) 6= Pρ′(λ), ρ 6= ρ′, ρ, ρ′ ∈ 1, . . . , r. For the degrees of the polynomialsoccurring in (2.80) holds
r∑
ρ=1
mρ · degree(Pρ(λ)) = degree(P (λ)) .
2. Furthermore, if P (λ) ∈ O(X)[λ], then the functions Pρ(λ) in the factorization (2.80),which a priori are in M(X)[λ] actually are in O(X)[λ] and therefore normalized andirreducible (ρ ∈ 1, . . . , r).
Proof: See [Bau].
34 J. Rama
Remark 2.14 Since M(X) is a field (see [Fo, 1.16 Remark]), it is in particular afactorial ring. This proves the first statement of Theorem 2.13. But the ring O(X)is not factorial; see [Re, 4* §2, p.94]. Thus to prove the Theorem 2.13, 2. needsmore than a standard algebraic argument. In Appendix E we give some notes on theinteresting algebraic structure of O(X).
Now by Theorem 2.13 the characteristic equation (2.79) is equivalent to
χ(λ) =
r∏
ρ=1
χρ(λ)mρ = 0 (2.81)
for some r ≤ N and some mρ ∈ N with χρ(λ) ∈ O(X)[λ] normalized and irreducible,χρ(λ) 6= χρ′(λ), ρ 6= ρ′, ρ, ρ′ ∈ 1, . . . , r. This factorization (2.81) is uniquely determined.If
nρ := degree(χρ(λ)) , (2.82)
it holds
r∑
ρ=1
mρ · nρ = dimRanΠ0 = N . (2.83)
Applying Theorem C.2 together with Corollary C.4 to each of the χρ(λ) (ρ ∈ 1, . . . , r)in the unique factorization (2.81) yields that there exist a Riemann surface Yρ, a branchednρ-sheeted holomorphic covering map πρ : Yρ → X and an analytic function Fρ ∈ O(Yρ),such that (π∗
ρχρ)(Fρ) = 0 (ρ ∈ 1, . . . , r). The algebraic function 〈Yρ, πρ, Fρ〉 defined by χρ
is uniquely determined (ρ ∈ 1, . . . , r).
Now fix some ρ ∈ 1, . . . , r. Using the ”localization”
χρ(λ) := L0(χρ(λ)) ∈ Cκ− 0[λ]
(which is defined as in (C.8) through (C.9) with ψ the identity map on X) gives in analogyto (C.10) the unique factorization
χρ(λ) =s∏
l=1
χρ,l(λ)ml (2.84)
for some ml ∈ N and some s ≤ nρ with χρ,l(λ) ∈ Cκ − 0[λ] irreducible and normalized,χρ,l(λ) 6= χρ,l′(λ), l 6= l′, l, l′ ∈ 1, . . . , s. If
nl := degree(χρ,l(λ)) , (2.85)
one has
s∑
l=1
ml · nl = nρ . (2.86)
As in (C.13) and (C.14), define
Dερ(0) := Uερ
(0)\κ ∈ Uερ(0) |Reκ ≤ 0, Imκ = 0 , (2.87)
where
Uερ(0) := κ ∈ X | |κ− 0| < ερ (2.88)
for some ερ > 0 small enough such that Uερ(0) contains no other critical values of πρ than
κ = 0, if κ = 0 is a critical value of πρ (respectively for some ερ > 0 small enough such that
Time Evolution of Quantum Resonance States 35
Uερ(0) contains no critical value of πρ, if κ = 0 is not a critical value of πρ). According to
Remark 2.16 Note that in our notation l = l(ρ) and i = i(l) = i(l(ρ)).
36 J. Rama
Remark 2.17 Each index l describes an nl-valued analytic function (correspondingto the polynomial χρ in the unique factorization (2.81)), and each eigenvalue λρ,l,i(κ)is a branch of this nl-valued analytic function.For fixed ρ and l the functions λρ,l,i(κ)nl
i=1, κ ∈ Dε(0), form the l-th cycle at pointκ = 0. (Cf. Remark C.14 and Remark C.17.) All eigenvalues λρ,l,i(κ)nl
i=1, κ ∈ Dε(0),of the l-th cycle have the same algebraic multiplicity
m(λρ,l,i(κ)) = ml ·mρ . (2.96)
Remark 2.18 Whenever there are (more than one) resonance eigenvalues λρ,l,i(κ)for κ 6= 0 in some neighborhood of κ = 0 splitting from the unperturbed eigenvalueλ0 = λρ,l,i(0), we call the point κ = 0 an exceptional point.
Originally, in [K] an exceptional point κ0 is defined to be a point in X (X some openconnected subset of C), where there is a splitting of λ0 (λ0 an eigenvalue of h(κ0))under perturbation. That is, however, at least two different eigenvalues of h(κ) forκ 6= κ0 in some neighborhood of κ0 must coincide at κ = κ0. Thus there is a splittingat and only at exceptional points; [K, II § 1.2, Remark 1.3]. Translating this into ourmore geometrical concept gives:
Assume the factorization (2.81) and the nρ-sheeted holomorphic coverings πρ describedafter (2.83). Let κ0 ∈ X. The point κ0 is exceptional, if and only if at least two differenteigenvalues λρ,l,i(κ) 6= λρ′,l′,i′(κ) for κ 6= κ0 in some neighborhood of κ0 coincide atκ = κ0, i.e, λρ,l,i(κ0) = λρ′,l′,i′(κ0).This is the case if κ0 is a critical value of πρ for at least one ρ ∈ 1, . . . , r. Or if in thefactorization (2.81) there are at least two distinct (i.e., ρ 6= ρ′) irreducible polynomialsχρ(λ) 6= χρ′(λ), both of degree 1, which coincide at κ0. That is: χρ(κ0, λ) = χρ′(κ0, λ),but χρ(κ, λ) 6= χρ′(κ, λ) for all κ 6= κ0, κ in some neighborhood of κ0.
Based on a Jordan decomposition, the following Proposition precisely shows the contribu-tions of the eigenvalues, the eigenprojections and eigennilpotents of h(κ) to the dynamicse−ith(κ) for t ≥ 0:
Proposition 2.19 Let V be a finite-dimensional C-linear space, dimV =: N < ∞.Let X ⊂ C be an open connected subset, 0 ∈ X. Let T (κ) (κ ∈ X) be a familyof bounded-analytic endomorphisms on V . Assume the factorization (2.81) of thecharacteristic polynomial χ(λ) ∈ O(X)[λ], χ(κ, λ) := det(T (κ) − λ) (κ ∈ X). LetUε(0) and Dε(0) be given by (2.91) and assume the factorization (2.94) of χ(κ, λ),κ ∈ Dε(0). Remember that l = l(ρ) and thus nl = nl(ρ) (see Remark 2.16). Let λρ,l,i(κ)(κ ∈ Dε(0)) denote the
∑rρ=1
∑sl=1 nl pairwise distinct eigenvalues (given by (2.90)
respectively by Proposition 2.15) of T (κ) with algebraic multiplicities m(λρ,l,i(κ)) =mρ ·ml (see Remark 2.17). Then:
1. For κ ∈ Dε(0), ρ ∈ 1, . . . , r, l ∈ 1, . . . , s and i ∈ 1, . . . , nl the eigenprojec-tions corresponding to λρ,l,i(κ) are the Riesz-projections
Πρ,l,i(κ) := − 1
2πi
∮
Γρ,l,i(κ)
(T (κ) − z)−1 dz , (2.97)
where Γρ,l,i(κ) is some curve in the resolvent set ρ(T (κ)), enclosing λρ,l,i(κ) butno other eigenvalue of T (κ) different from λρ,l,i(κ).
2. For κ ∈ Dε(0), ρ ∈ 1, . . . , r, l ∈ 1, . . . , s and i ∈ 1, . . . , nl the eigennilpo-tents corresponding to λρ,l,i(κ) are given by
Proof of Proposition 2.19: (2.97) and (2.98) hold according to (B.1) and (B.2). The state-ments (2.99) - (2.101) are proven as follows: Let t ∈ R. In Lemma B.1 choose φ(z) := e−itz,z ∈ C. Then φ(k)(z) = (−it)k e−itz (z ∈ C, k ∈ N). Applying this to the setting of Propo-sition 2.19 finishes the proof.
We will now analyze in more detail the structure of the eigenvalues, eigenprojections andeigennilpotents for the analytic family h(κ).
Definition 2.20 Let kl ∈ N ∪ ∞ and let λρ,l,i(κ) be as in (2.95). Then
λρ,l,i(κ) branches in generation kl :⇔
λρ,l,i(κ) = a(ρ,l,i)0 + a(ρ,l,i)
nlκ+ a
(ρ,l,i)2nl
κ2 + . . .+ a(ρ,l,i)kl·nl
κkl +
∞∑
ν=1
a(ρ,l,i)kl·nl+ν κ
kl+ν/nl
with a(ρ,l,i)kl·nl+ν 6= 0 for at least one ν, 1 ≤ ν < nl.
Note that there is no branching of λρ,l,i(κ) (κ ∈ Dε(0)), if and only if kl = ∞.
Remark 2.21 At this point we derive an additional important structure of h(κ):
From (2.42) and (2.78) follows by comparing coefficients
h(0) = λ0Π0 , h(1) = Π0VΠ0 .
Obviously, h(0) is a symmetric endomorphism on RanΠ0. h(1) is a symmetric endo-morphism on RanΠ0, since V is symmetric (see Remark 2.4). Since
[h(0), h(1)] = λ0[Π0,Π0VΠ0] = 0 ,
the operators h(0) and h(1) are simultaneously diagonable (in a basis of common eigen-vectors). Note that λ0 and the eigenvalues of h(1) are real.
Lemma 2.22 Assume (A1) - (A6). Let h(κ) be the analytic matrix family given by(2.41) and (2.78). Assume the characteristic equation (2.94) of h(κ). Assume theresults of Proposition 2.15. Then, as a consequence of h(0) and h(1) being semisimple(by Remark 2.21), one has:
where λ(1)(ρ,l,i) denotes an eigenvalue of h(1) making the contribution to λρ,l,i(κ) and
λ(2)(ρ,l,i) an eigenvalue of h(2) making the contribution to λρ,l,i(κ). Furthermore,
a(ρ,l,i)ν = 0 (2.105)
for ν < 2 · nl with ν 6= 0 and ν 6= nl.
Proof: The proof is given by the reduction process described in [K, II § 2.3]; see in par-ticular [K, II § 2.3, Theorem 2.3] and [K, II § 2.3, (2.41)]. This reduction process can beapplied to the setting of Lemma 2.22, since h(0) and h(1) are semisimple.
Thus by combining (2.102) - (2.105) with (2.90) eigenvalues of h(κ) have the form
λρ,l,i(κ) = λ0 + λ(1)(ρ,l,i)κ+ λ
(2)(ρ,l,i)κ
2 +∞∑
ν=1
a(ρ,l,i)2·nl+ν κ
2+ν/nl (κ ∈ Dε(0)) (2.106)
for some a(ρ,l,i)2nl+ν ∈ C. We remark that generically a branching of λρ,l,i(κ) in each generation
≥ 2 is possible.
For an abbreviation we introduce the (kl, 0)-Jet
J(kl,0)ρ,l,i (κ) := λ0 + λ
(1)(ρ,l,i)κ+ λ
(2)(ρ,l,i)κ
2 + a(ρ,l,i)3·nl
κ3 + . . .+ a(ρ,l,i)kl·nl
κkl (κ ∈ Dε(0)) (2.107)
of λρ,l,i(κ) for some kl ≥ 2. This is the part of λρ,l,i(κ), where no fractional exponents arise;cf. (2.106). And we define the (kl,∞)-Jet
J(kl,∞)ρ,l,i (κ) :=
∞∑
ν=1
a(ρ,l,i)kl·nl+ν κ
kl+ν/nl (κ ∈ Dε(0)) (2.108)
of λρ,l,i(κ), i.e., the part which consists fractional exponents. In particular
λρ,l,i(κ) = J(kl,0)ρ,l,i (κ) + J
(kl,∞)ρ,l,i (κ) (2.109)
for κ ∈ Dε(0) and some kl ≥ 2.
Lemma 2.23 Let λρ,l,i(κ) be as in (2.109) with ρ ∈ 1, . . . , r fixed and l ∈ 1, . . . , sfixed. Then it holds
Proof: Under analytic continuation around κ = 0 the branches λρ,l,i(κ) (κ ∈ Dε,
i ∈ 1, . . . , nl) transform one into another; see Remark C.14. But J(kl,0)ρ,l,i (κ) (κ ∈ Dε(0),
i ∈ 1, . . . , nl) is a polynomial in κ of degree kl, thus an analytic function of κ. Thus itdoes not change under analytic continuation.
Remark 2.24 Lemma 2.23 allows to define
J(kl,0)ρ,l (κ) := J
(kl,0)ρ,l,i (κ) (κ ∈ Dε(0), i ∈ 1, . . . , nl) .
So in (2.107) and (2.109) the jets J(kl,0)ρ,l,i (κ) can be rephrased by J
(kl,0)ρ,l (κ):
λρ,l,i(κ) = J(kl,0)ρ,l (κ) + J
(kl,∞)ρ,l,i (κ) , (2.111)
J(kl,0)ρ,l (κ) = λ0 + λ
(1)(ρ,l)κ+ λ
(2)(ρ,l)κ
2 + a(ρ,l)3nl
κ3 + . . .+ a(ρ,l)kl·nl
κkl (2.112)
Time Evolution of Quantum Resonance States 39
for some kl ≥ 2 and all κ ∈ Dε(0), i ∈ 1, . . . , nl. Remember that λ0 ∈ R and
λ(1)(ρ,l) ∈ R by Remark 2.21.
This motivates the following
Definition 2.25 Assume (2.107) - (2.109). Let ρ ∈ 1, . . . , r and l ∈ 1, . . . , s both
be fixed. The multiplicity of the jet J(kl,0)ρ,l,i (κ) is defined to be the number nl; in
symbols:
m(J(kl,0)ρ,l,i (κ)) = nl (κ ∈ Dε(0)) .
Remark 2.26 Remember that nl has been defined as the degree of χρ,l(λ); see (2.85).And nl coincides with the number of sheets introduced in Appendix C.2; see in par-ticular Remark C.13 respectively Remark C.16.
For an overview we give
Lemma 2.27 Assume (2.107) - (2.109). Let ρ ∈ 1, . . . , r and l ∈ 1, . . . , s both befixed. Then the following statements are equivalent:
(1) m(J(kl,0)ρ,l,i (κ)) = nl = 1 (κ ∈ Dε(0)).
(2) λρ,l,i(κ), κ ∈ Dε(0), forms a 1-cycle.
(3) kl = ∞.
(4) λρ,l,i(·) : Dε(0) → C extends to an analytic function λρ,l,i(·) : Uε(0) → C.
(5) There is no branching of λρ,l,i(κ) (κ ∈ Dε(0)).
Proof:(1)⇔(2): See Remark C.14 and Remark C.17.(4)⇔(3)⇔(5): See Definition 2.20.(1)⇒(3): This is obvious by inserting nl = 1 into Definition 2.20.
(1)⇐(3): Let kl = ∞. Then J(kl=∞,∞)ρ,l,i (κ) = 0 (i ∈ 1, . . . , nl, κ ∈ Dε(0)), and by use of
which are analytic functions (polynomials) in κ. Thus (1) – and (4) – follows, since λρ,l,i(κ)and λρ,l,i′(κ) do not change under analytic continuation around κ = 0 but also transformone into another under analytic continuation around κ = 0.
Remark 2.28 At this point we give an useful estimate for e−itλρ,l,i(κ), which we willneed later: For ρ ∈ 1, . . . , r, l ∈ 1, . . . , s, i ∈ 1, . . . , nl inequality (2.60) togetherwith Remark 2.24 gives
Imλρ,l,i(κ) = −|Imλρ,l,i(κ)| =
−κ2 |Imλ(2)(ρ,l)| +O(κ2+1/nl) , kl = 2
−κ2 |Imλ(2)(ρ,l)| +O(κ3) , kl > 2
(2.113)
and
Reλρ,l,i(κ) =
λ0 + κλ(1)(ρ,l) + κ2 Reλ
(2)(ρ,l) +O(κ2+1/nl) , kl = 2
λ0 + κλ(1)(ρ,l) + κ2 Reλ
(2)(ρ,l) +O(κ3) , kl > 2
40 J. Rama
as κ → 0 for real κ ≥ 0 small enough. Thus for ρ ∈ 1, . . . , r, l ∈ 1, . . . , s,i ∈ 1, . . . , nl and 0 ≤ t <∞
Now we have analyzed in detail the form of eigenvalues of the analytic matrix family h(κ).Knowing the form of eigenvalues allows to draw conclusions about the form of the corre-sponding eigenprojections and eigennilpotents. Results in this direction are formulated inthe following Theorems, freely quoted from [K]:
Theorem 2.29 [K, II § 1.5, Theorem 1.8]
Let X ⊂ C be an open connected subset, let V be a finite-dimensional C-linear space.Assume the analytic matrix family T (·) : X → End(V ), κ 7→ T (κ). Then:
1. Eigenvalues λρ,l,i(κ), eigenprojections Πρ,l,i(κ) and eigennilpotents Dρ,l,i(κ) for T (κ)are (branches of) analytic functions with only algebraic singularities at some (but notnecessarily all) exceptional points.
2. λρ,l,i(κ) and Πρ,l,i(κ) have all critical values (called ”branch points” in [K]; see Remark2.31 below) in common including the branching order (given in Definition D.22) of thebranch points corresponding to these critical values. If in particular λρ,l,i(κ) is single-valued near an exceptional point κ0 ∈ X (i.e., λρ,l,i(κ) constitutes a 1-cycle at κ0;see Remark C.14 and Remark C.17), then Πρ,l,i(κ) and Dρ,l,i(κ) are also single-valuedthere.
3. The critical values for λρ,l,i(κ) and Πρ,l,i(κ) may or may not be critical values forDρ,l,i(κ).
Remark 2.30 We remark that Πρ,l,i(κ) andDρ,l,i(κ) are matrix-valued analytic func-tions. Thus the statement of Theorem 2.29, 1. is equivalent to the matrix elementsof Πρ,l,i(κ) and Dρ,l,i(κ) being analytic functions with only algebraic singularities atsome (but not necessarily all) exceptional points.
Remark 2.31 These points in X, which we are calling ”critical values”, are called”branch points” in Kato’s book [K] (since in [K] the concept of Riemann surfaces ismissing). So our definition of a branch point differs from Kato’s definition of a branchpoint.
Theorem 2.32 (Butler) [K, II § 1.6, Theorem 1.9]
Assume the same notations and conditions as in Theorem 2.29. Let κ0 ∈ X. If κ0 = κ
is a critical value of λρ,l,i(κ) (and therefore also of Πρ,l,i(κ)) of order nl − 1 ≥ 1,then Πρ,l,i(κ) has a pole there; that is, the Laurent expansion of Πρ,l,i(κ) in powers of(κ − κ0)
2.2.3 Factorizing the Characteristic Polynomial of h(κ)
Of course, χ(λ) ∈ O(X)[λ] is either reducible or irreducible. First assume χ(λ) ∈ O(X)[λ]is irreducible. This corresponds to r = ρ = 1 in the unique factorization (2.81). Thusone has χ(λ) = χ1(λ). Clearly, degree(χ1(λ)) = N . Then the corresponding polynomialχ(λ) = χ1(λ) ∈ Cκ−0[λ] of degree N introduced in (2.84) is itself irreducible or factorizesuniquely into
χ(λ) = χ1(λ) =
s∏
l=1
χ1,l(λ)ml (2.115)
for some s ≤ N , χ1,l(λ) ∈ Cκ − 0 irreducible and normalized, χ1,l(λ) 6= χ1,l′(λ), l 6= l′,l, l′ ∈ 1, . . . , s. If degree(χ1,l(λ)) =: nl, then
s∑
l=1
ml · nl = N .
Applying Theorem C.2 together with Corollary C.4 to χ1(λ) yields that there exists aRiemann surface Y1, a branched N -sheeted holomorphic covering map π1 : Y → X and ananalytic function F1 ∈ O(Y ), such that (π∗
1χ1)(F1) = 0. The algebraic function 〈Y1, π1, F1〉defined by χ = χ1 is uniquely determined.
We start with the simple case of s = l = ml = 1 in the factorization (2.115). That is,χ(λ) = χ1(λ) = χ1,1(λ) ∈ Cκ − 0[λ] is itself irreducible of degree N . Then κ = 0 is acritical value of π1.
Remark 2.33 The case κ = 0 being not a critical value of π1 can be excluded by thefollowing arguments:
Suppose κ = 0 is not a critical value of π1. Then λ1,1,i(0) 6= λ1,1,i′(0), i 6= i′ (i, i′ ∈1, . . . , N). But this contradicts λ1,1,i(0) = λ0 (i ∈ 1, . . . , N), which follows fromour starting-assumption that λ0 is an eigenvalue of h(0) with algebraic multiplicitym(λ0) = N ; see (2.77) and (A4).
Now let Uε(0) := Uε1(0) and Dε(0) := Dε1
(0) be defined as in (2.87) and (2.88) with someε := ε1 > 0 small enough such that Uε(0) contains no other critical value of π than κ = 0.Solutions of the characteristic equation
As pointed out in Remark C.14 and Remark C.15, solutions of (2.116) describe the case,where the eigenvalue λ0 of h0 splits under perturbation into one N -cycle
σ(h(κ)) =λ1,1,1(κ), λ1,1,2(κ), . . . , λ1,1,N (κ)
(κ ∈ Dε(0)) , (2.120)
where all the eigenvalues λ1,1,i(κ) have algebraic multiplicity
m(λ1,1,i(κ)) = 1 (i ∈ 1, . . . , N , κ ∈ Dε(0)) . (2.121)
42 J. Rama
Thus from (B.3) follows that all corresponding eigennilpotents D1,1,i(κ) vanish:
D1,1,i(κ) = 0 (i ∈ 1, . . . , N , κ ∈ Dε(0)) . (2.122)
Now by use of Proposition 2.19, 3. and (2.122) one gets
e−ith(κ) =N∑
i=1
e−itλ1,1,i(κ)Π1,1,i(κ)
= e−itJ(k1,0)1,1 (κ)
N∑
i=1
e−itJ(k1,∞)1,1,i (κ)Π1,1,i(κ) (κ ∈ Dε(0), t ∈ R) . (2.123)
According to Theorem 2.32 the eigenprojections Π1,1,i(κ) (i ∈ 1, . . . , N, κ ∈ Dε(0)) allhave a pole at κ = 0. At this point one ought to determine the order of these poles. (ForN=2 we shall do this in Lemma 2.35.)
As one can easily see, for arbitrary N < ∞ the analysis of e−ith(κ) becomes more compli-cated for χ(λ) = χ1(λ) reducible in Cκ − 0[λ] or even in the case if χ(λ) ∈ O(X)[λ] isitself reducible. In fact this turned out to be a (hard) combinatorial problem, which is inpreparation.
For these reasons we restrict our analysis of e−ith(κ) to the case N = dim RanΠ0 = 2:
2.2.4 Factorizations in the Special Case of dim RanΠ0 = 2 and χ(λ)reducible
Let dimRanΠ0 = 2 and assume χ(λ) ∈ O(X)[λ] is reducible. Then by Theorem 2.13, 2.there exists a unique factorization of χ(λ) into irreducible polynomials in O(X)[λ]. Thisunique factorization is either:
Case (1): χ(λ) = χ1(λ) χ2(λ)
with χ1(λ) 6= χ2(λ) and χρ(λ) ∈ O(X)[λ] a normalized and irreducible polynomial ofdegree nρ = 1 (ρ ∈ 1, 2).
Or:
Case (2): χ(λ) = χ1(λ)2
with χ1(λ) ∈ O(X)[λ] a normalized and irreducible polynomial of degree n1 = 1.
In both cases – case (1) and case (2) – applying Theorem C.2 together with Corollary C.4to χρ(λ) (ρ ∈ 1, 2) yields:
There exists a Riemann surface Yρ, a ”branched” (nρ = 1)-sheeted holomorphic coveringmap πρ : Yρ → X and an analytic function Fρ ∈ O(Y ), such that (π∗
ρχρ)(Fρ) = 0. Thealgebraic function 〈Yρ, πρ, Fρ〉 defined by χρ is uniquely determined.
So the eigenvalue λ0 of h0 splits under perturbation into two 1-cycles. More precisely:
σ(h(κ)) =λ1,1,1(κ), λ2,1,1(κ)
(κ ∈ Uε(0) :=
2⋂
ρ=1
Uερ(0)) ,
where by Remark 2.24
λρ,1,1(κ) = J(∞,0)ρ,1 (κ) = λ0 + λ
(1)(ρ,1)κ+ λ
(2)(ρ,1)κ
2 +∞∑
ν=3
a(ρ,1)ν κν
for ρ ∈ 1, 2, κ ∈ Uερ(0) with algebraic multiplicities
m(λρ,1,1(κ)) = 1 (ρ ∈ 1, 2, κ ∈ Uερ(0)) .
As we have seen, the eigenvalues λρ,1,1(κ) of h(κ) are single-valued analytic functions fromUερ
(0) to C. Thus by Theorem 2.29 their corresponding eigenprojections Πρ,1,1(κ) are alsosingle-valued and analytic there:
Πρ,1,1(κ) =
∞∑
ν=0
Π(ν)(ρ,1,1)κ
ν = Π(0)(ρ,1,1) +
∞∑
ν=1
Π(ν)(ρ,1,1)κ
ν = Π(0)(ρ,1,1) +O(κ) (|κ| → 0) (2.124)
for some Π(ν)(ρ,1,1) ∈ End(RanΠ0) and all ρ ∈ 1, 2, κ ∈ Uερ
(0). Since both of the eigenvalues
have algebraic multiplicity one, the corresponding eigennilpotents are exactly equal zero:
Dρ,1,1(κ) = 0 (ρ ∈ 1, 2 , κ ∈ Uερ(0)) .
Now Proposition 2.19 leads to
e−ith(κ) =
2∑
ρ=1
e−itλρ,1,1(κ)Πρ,1,1(κ) (κ ∈ Uε(0)) . (2.125)
Then using Remark 2.28 and (2.124) in (2.125) yields for 0 ≤ κ < ε and t ∈ R
e−ith(κ) = e−itλ0
2∑
ρ=1
e−it
(κλ
(1)
(ρ,1)+κ2Reλ
(2)
(ρ,1)+ReO(κ3)
)
e−tκ2|Imλ
(2)
(ρ,1)|+t·ImO(κ3)(Π
(0)(ρ,1,1) +O(κ)
)
as κ→ 0. This gives the estimate
‖e−ith(κ)‖ ≤2∑
ρ=1
e−tκ2|Imλ
(2)
(ρ,1)|+t·ImO(κ3)(
1 +O(κ))
≤ 2 e−tκ2 min
ρ∈1,2|Imλ
(2)
(ρ,1)|+t·ImO(κ3)(
1 +O(κ))
(κ→ 0) (2.126)
for 0 ≤ κ < ε and t ≥ 0.
Remark 2.34 Let ∞ > dim RanΠ0 =: N be arbitrary. If the characteristic poly-nomial χ(λ) ∈ O(X)[λ] of h(κ) factorizes into N irreducible polynomials (of degreeone) in O(X)[λ],
χ(λ) =
N∏
ρ=1
χρ(λ) ,
then analogously to (2.126) one gets the estimate
‖e−ith(κ)‖ ≤N∑
ρ=1
e−tκ2|Imλ
(2)
(ρ,1)|+t·ImO(κ3)(1 +O(κ)
)
≤ N e−tκ2 min
ρ∈1,...,N|Imλ
(2)
(ρ,1)|+t·ImO(κ3)(
1 +O(κ))
(κ→ 0)
for 0 ≤ κ < ε and t ∈ R.
44 J. Rama
Analyzing Case (2)
In case (2) one has
χ(λ) := L0(χ1(λ))2 = χ1(λ)2 = χ1,1(λ)2
with
χ1,1(λ) ∈ Cκ− 0[λ]
irreducible and normalized and
degree(χ1,1,(λ)) = 1 .
Case (2) corresponds to ”no splitting” of λ0, i.e., permanent degeneracy. In this case wehave for κ ∈ Uε1
(0) = Uε(0)
σ(h(κ)) = λ1,1,1(κ) ,
λ1,1,1(κ) = J(∞,0)1,1 (κ)
(2.112)= λ0 + λ
(1)(1,1)κ+ λ
(2)(1,1)κ
2 +∞∑
ν=3
a(1,1)ν κν , (2.127)
m(λ1,1,1(κ)) = 2 .
Here λ1,1,1(·) : Uε(0) → C is again single-valued analytic. So, according to Theorem 2.29,the corresponding eigenprojection and eigennilpotent are also single-valued analytic. Theeigenprojection is given by
Π1,1,1(κ) = Π0 +∞∑
ν=1
Π(ν)(1,1,1)κ
ν = Π0 +O(|κ|) (|κ| → 0) (2.128)
for some Π(ν)(1,1,1) ∈ End(RanΠ0) and all κ ∈ Uε(0). The corresponding eigennilpotent is
given by
D1,1,1(κ) =(h(κ) − λ1,1,1(κ)
)Π1,1,1(κ) (κ ∈ Uε(0)) ; (2.129)
see Proposition 2.19. Since Π1,1,1(κ) ∈ End(RanΠ0), it holds Π1,1,1(κ) = Π0Π1,1,1(κ)Π0.By use of Remark 2.21 and (2.127) one obtains
(h(κ) − λ1,1,1(κ)
)Π0 =
((h0 + κh(1) +O(|κ|2)
)−
(λ0 + κλ
(1)(1,1) +O(|κ|2)
))
Π0
=((λ0 + κλ
(1)(1,1) +O(|κ|2)
)−
(λ0 + κλ
(1)(1,1) +O(|κ|2)
))
Π0
= O(|κ|2) (|κ| → 0) .
Thus the eigennilpotent is
D1,1,1(κ) =(h(κ) − λ1,1,1(κ)
)Π0Π1,1,1(κ)Π0 =:
∞∑
ν=2
D(ν)(1,1,1)κ
ν = O(|κ|2) (2.130)
as |κ| → 0 for all κ ∈ Uε(0) and some D(ν)(1,1,1) ∈ End(RanΠ0). Proposition 2.19 yields
e−ith(κ) = e−itλ1,1,1(κ)Π1,1,1(κ) +D1,1,1(κ)′
= e−itλ1,1,1(κ)Π1,1,1(κ) − it e−itλ1,1,1(κ)D1,1,1(κ)
= e−itλ1,1,1(κ)(1 − itD1,1,1(κ)
)Π1,1,1(κ) (κ ∈ Uε(0) , t ∈ R) . (2.131)
Then inserting (2.128) and (2.130) into (2.131) gives
as an operator on RanΠ0 for all κ ∈ Uε(0) and t ∈ R. Further, for all t ∈ R and real0 ≤ κ < ε one obtains by use of Remark 2.28
e−itλ1,1,1(κ) = e−it
(λ0+κλ
(1)
(1,1)+κ2Reλ
(2)
(1,1)+ReO(κ3)
)
e−tκ2|Imλ
(2)
(1,1)|+t·ImO(κ3) (2.134)
as κ→ 0. Thus combining (2.133) and (2.134) gives
e−ith(κ) = e−it
(λ0+κλ
(1)
(1,1)+κ2Reλ
(2)
(1,1)+ReO(κ3)
)
e−tκ2|Imλ
(2)
(1,1)|+t·ImO(κ3) ·
·(Π0 +O(κ) − itO(κ2) − itO(κ3)
),
‖e−ith(κ)‖ ≤ e−tκ2|Imλ
(2)
(1,1)|+t·ImO(κ3) (
1 +O(κ) +O(κ2t) +O(κ3t))
(2.135)
as κ→ 0 for all 0 ≤ κ < ε and t ∈ R.
2.2.5 Factorizations in the Special Case of dim RanΠ0 = 2 and χ(λ)irreducible
Let dim RanΠ0 = 2 and assume χ(λ) is irreducible in O(X)[λ]. Then
χ(λ) := L0(χ(λ))
is either irreducible or reducible in Cκ− 0[λ].
Let χ(λ) be irreducible in Cκ− 0[λ]. This corresponds to
Case (3): χ(λ) = χ1(λ) = χ1,1(λ) irreducible in Cκ− 0[λ],
degree(χ1,1(λ)) = 2.
Now assume χ(λ) is reducible in Cκ − 0[λ] , but therefore uniquely factorizing into aproduct of irreducible polynomials in Cκ− 0[λ]. This unique factorization is either
Case (4): χ(λ) = χ1(λ) = χ1,1(λ) χ1,2(λ)
with χ1,1(λ) 6= χ1,2(λ), χ1,l(λ) irreducible in Cκ−0[λ] and degree(χ1,l(λ)) = nl = 1(l ∈ 1, 2).
or
Case (5): χ(λ) = χ1(λ) = χ1,1(λ)2
with χ1,1(λ) irreducible in Cκ− 0[λ] and degree(χ1,1(λ)) = nl=1 = 1.
Analyzing Case (3)
In analogy to (2.117) - (2.122) with N = 2 we obtain
σ(h(κ)) =λ1,1,1(κ), λ1,1,2(κ)
(κ ∈ Dε(0)) ,
λ1,1,i(κ) = J(k1,0)1,1 (κ) + J
(k1,∞)1,1,i (κ) (i ∈ 1, 2, κ ∈ Dε(0)) , (2.136)
m(λ1,1,i(κ)) = 1 (i ∈ 1, 2, κ ∈ Dε(0))
and
D1,1,i(κ) = 0 (i ∈ 1, 2, κ ∈ Dε(0)) . (2.137)
We abbreviate k := k1. Now using (2.136) - (2.137) in Proposition 2.19, 3. yields
e−ith(κ) =2∑
i=1
e−itλ1,1,i(κ)Π1,1,i(κ) (2.138)
= e−itJ(k,0)1,1 (κ)
2∑
i=1
e−itJ(k,∞)1,1,i (κ)Π1,1,i(κ) (2.139)
46 J. Rama
for t ∈ R and κ ∈ Dε(0). Depending on the branching generation k (see Definition 2.20), itholds
e−itJ(k,0)1,1 (κ) =
e−it
(λ0+κλ
(1)
(1,1)+κ2Reλ
(2)
(1,1)
)
e−tκ2|Imλ
(2)
(1,1)|
, k = 2
e−it
(λ0+κλ
(1)
(1,1)+κ2Reλ
(2)
(1,1)+ReO(κ3)
)
e−tκ2|Imλ
(2)
(1,1)|+t·ImO(κ3)
, k > 2(2.140)
as κ→ 0 for all t ∈ R and 0 < κ < ε. Further it holds
e−itJ(k,∞)1,1,i (κ) = e−itO(κk+1/2) (κ→ 0 , t ∈ R , 0 < κ < ε) . (2.141)
The branching order of λ1,1,i(·) at κ = 0 (in the sense of Definition D.22) is
b(π, 0) = m(π, 0) − 1 = n1 − 1 = 2 − 1 = 1 .
Now by Butler’s Theorem (Theorem 2.32) the corresponding eigenprojections Π1,1,i(·) (i ∈1, 2) have a pole at κ = 0. That is, the Laurent expansion of Π1,1,i(κ) in powers of κ1/2
contains negative powers. The following Lemma determines the order of this pole:
Lemma 2.35 Assume (2.136) for some k := k1 ≥ 2. Then the corresponding eigen-projections Π1,1,i(κ) (i ∈ 1, 2, κ ∈ Dε(0)) have a pole of order k + 1 + 1
2 − 2k atκ = 0. That is
Π1,1,i(κ) = O(|κ|k+1+ 12−2k) (|κ| → 0) (2.142)
for κ ∈ Dε(0) and i ∈ 1, 2.
To prove Lemma 2.35 we will need
Lemma 2.36 ([K, I § 4.2, (4.12)])
Let V be a finite-dimensional linear space with dimV := N < ∞. Let T ∈ End(V )and assume T−1 exists. Then
‖T−1‖ ≤ C‖T‖N−1
|detT | , (2.143)
where C is a constant independent of T but depending on the norm employed. If V isan unitary space, one can set C = 1.
Proof of Lemma 2.35: We construct separating contours in X for λ1,1,1(κ) and λ1,1,2(κ):For κ ∈ Dε(0) it holds
dist(λ1,1,1(κ), λ1,1,2(κ)
)=
∣∣λ1,1,1(κ) − λ1,1,2(κ)
∣∣
(2.111)=
∣∣∣J
(k,0)1,1 (κ) + J
(k,∞)1,1,1 (κ) − J
(k,0)1,1 − J
(k,∞)1,1,2 (κ)
∣∣∣
=∣∣∣J
(k,∞)1,1,1 (κ) − J
(k,∞)1,1,2 (κ)
∣∣∣
(2.108)=
∣∣∣∣∣
∞∑
ν=1
a(1,1,1)2k+ν κk+ν/2 −
∞∑
ν=1
a(1,1,2)2k+ν κk+ν/2
∣∣∣∣∣
=
∣∣∣∣∣
∞∑
ν=1
(a(1,1,1)2k+ν − a
(1,1,2)2k+ν
)κk+ν/2
∣∣∣∣∣
= O(|κ|k+1/2) = o(|κ|k) (|κ| → 0) .
Let
0 < ρ <|a(1,1,1)
2k+1 − a(1,1,2)2k+1 |
2
Time Evolution of Quantum Resonance States 47
and
∂Kρ(a(1,1,i)2k+1 ) :=
z ∈ C
∣∣ |z − a
(1,1,i)2k+1 | = ρ
(i ∈ 1, 2) .
Let S1 :=z ∈ C
∣∣ |z| = 1
, s := ρ · s (s ∈ S1). Then a proper parameterization of the
separating contour is
∂Kρ(a(1,1,i)2k+1 ) → Γ1,1,i(κ)
s 7→ J(k,0)1,1 (κ) + a
(1,1,i)2k+1 κ
k+1/2 + sκk+1/2 =: zi(s) (2.144)
for i ∈ 1, 2 and κ ∈ Dε(0). By use of this parameterization (2.144), the spectral projectionsare given by
Π1,1,i(κ) := − 1
2πi
∮
Γ1,1,i(κ)
(h(κ)− z)−1 dz
= − 1
2πi
∮
∂Kρ(a(1,1,i)2k+1 )
(h(κ) − zi(s))−1zi
′(s) ds (2.145)
for i ∈ 1, 2 and κ ∈ Dε(0). Equation (2.144) yields
zi′(s) = κk+1/2 (i ∈ 1, 2, κ ∈ Dε(0)) . (2.146)
Thus by inserting (2.146) into (2.145) we get
Π1,1,i(κ) = − κk+1/2
2πi
∮
∂Kρ(a(1,1,i)2k+1 )
(h(κ) − zi(s)
)−1ds (2.147)
for i ∈ 1, 2 and κ ∈ Dε(0). By use of a further parameterization
∂Kρ(a(1,1,i)2k+1 ) : [0, 2π) → C
φ 7→ s(φ) := ρ eiφ + a(1,1,i)2k+1
in (2.147) we then get the estimate
‖Π1,1,i(κ)‖ =∥∥∥ − κk+1/2
2πi
2π∫
0
(h(κ) − zi(s(φ))
)−1iρ eiφ dφ
∥∥∥
≤ ρ |κ|k+ 12 max
φ∈[0,2π)‖(h(κ) − zi(s(φ)))−1‖ (2.148)
for i ∈ 1, 2 and κ ∈ Dε(0). We will now estimate the resolvent on the r.h.s. of (2.148).Applying Lemma 2.36 to h(κ) − zi(s(φ)), using Cramer’s rule and taking into accountm(λ1,1,i′(κ)) = 1 (i′ ∈ 1, 2) yields
for i ∈ 1, 2 and κ ∈ Dε(0). Finally combining (2.155) with (2.148) proves (2.142).
Now Butler’s Theorem together with Lemma 2.35 give the Laurent expansion of Π1,1,i(κ):
Π1,1,i(κ) =
∞∑
ν=−2k+3
Π(ν)(1,1,i)κ
ν/2 (i ∈ 1, 2 , κ ∈ Dε(0)) (2.156)
for some Π(ν)(1,1,i) ∈ End(RanΠ0).
Finally inserting (2.140) and (2.141) into (2.139) and taking into account Lemma 2.35 yields
e−ith(κ) = e−it
(λ0+κλ
(1)
(1,1)+κ2Reλ
(2)
(1,1)+ReO(κ3)
)
e−tκ2|Imλ
(2)
(1,1)|+t·ImO(κ3)
e−itO(κk+1/2) ··O(κk+1+ 1
2−2k) (2.157)
as κ → 0 for all 0 < κ < ε, t ∈ R and some k ≥ 2. For the special case k = 2 this issimplified by (2.118) to
e−ith(κ) = e−it
(λ0+κλ
(1)
(1,1)+κ2Reλ
(2)
(1,1)
)
e−tκ2|Imλ
(2)
(1,1)|e−itO(κ2+1/2)O(κ−
12 ) (2.158)
as κ → 0 for all 0 < κ < ε and t ∈ R. So from (2.157) and (2.158) follows for some k ≥ 2and all t ∈ R
‖e−ith(κ)‖ = e−tκ2|Imλ
(2)
(1,1)|+t·Im o(κ2)
O(κk+1+ 12−2k) (2.159)
as κ→ 0 for all 0 < κ < ε.
Analyzing Case (4)
In analogy to the analysis of case (1) in Section 2.2.4 one obtains:
The eigenvalue λ0 of h0 splits under perturbation into two 1-cycles:
σ(h(κ)) =λ1,1,1(κ), λ1,2,1(κ)
(κ ∈ Uε(0) := Uε1
(0)) ,
Time Evolution of Quantum Resonance States 49
where
λ1,l,1(κ) = J(∞,0)1,l (κ) = λ0 + λ
(1)(1,l)κ+ λ
(2)(1,l)κ
2 +
∞∑
ν=3
a(1,l,1)ν κν
for l ∈ 1, 2, κ ∈ Uε(0) with algebraic multiplicities
m(λ1,l,1(κ)) = 1 (l ∈ 1, 2, κ ∈ Uε(0)) .
The eigenvalues λ1,l,1(κ) of h(κ) are single-valued analytic functions from Uε(0) to C. Thusby Theorem 2.29 their corresponding eigenprojections Π1,l,1(κ) are also single-valued andanalytic there:
Π1,l,1(κ) =
∞∑
ν=0
Π(ν)(1,l,1)κ
ν = Π(0)(1,l,1) +
∞∑
ν=1
Π(ν)(1,l,1)κ
ν = Π(0)(1,l,1) +O(κ) (|κ| → 0)
for some Π(ν)(1,l,1) ∈ End(RanΠ0) and all l ∈ 1, 2, κ ∈ Uε(0). Since both of the eigenvalues
have algebraic multiplicity one, the corresponding eigennilpotents are exactly equal zero:
D1,l,1(κ) = 0 (l ∈ 1, 2 , κ ∈ Uε(0)) .
finally this leads to
e−ith(κ) =
2∑
l=1
e−itλ1,l,1(κ)Π1,l,1(κ) (κ ∈ Uε(0)) (2.160)
and
e−ith(κ) = e−itλ0
2∑
l=1
e−it
(κλ
(1)
(1,l)+κ2Reλ
(2)
(1,l)+ReO(κ3)
)
e−tκ2|Imλ
(2)
(1,l)|+t·ImO(κ3)(Π
(0)(1,l,1) +O(κ)
)
as κ→ 0 for 0 ≤ κ < ε and t ∈ R. This gives the estimate
‖e−ith(κ)‖ ≤2∑
l=1
e−tκ2|Imλ
(2)
(1,l)|+t·ImO(κ3)(1 +O(κ)
)
≤ 2 e−tκ2 min
l∈1,2|Imλ
(2)
(1,l)|+t·ImO(κ3)(
1 +O(κ))
(κ→ 0) (2.161)
for 0 ≤ κ < ε and t ≥ 0.
Analyzing Case (5)
Case (5) is equivalent to case (2). So all results of Section 2.2.4 hold also in case (5).
50 J. Rama
2.3 Results
Proposition 2.37 Assume (A1) - (A6) with dimRanΠ0 = N = 2. Let h(κ) be theanalytic matrix family found in (2.41). Let χ(λ) ∈ O(X)[λ] denote the characteristicpolynomial of h(κ) introduced in (2.79) satisfying a factorization (2.81). Let χ(λ)denote its corresponding polynomial in Cκ− 0[λ] with a factorization (2.92).
Generically, the following factorizations of χ(λ) and χ(λ) can occur:
Then, depending on these factorizations, the λ0-group for h(κ) is characterized as follows:
Time Evolution of Quantum Resonance States 51
Proposition 2.38 Assume (A1) - (A6) with dim RanΠ0 = N = 2. Let h(κ) bethe analytic matrix family found in (2.41). Assume the results of Proposition 2.37.Depending on the factorizations (1) - (5) in Proposition 2.37 the λ0-group of h(κ) ischaracterized as follows:
In case (1) the eigenvalue λ0 of h0 splits under perturbation into two 1-cycles
σ(h(κ)) =λ1,1,1(κ), λ2,1,1(κ)
(κ ∈ Uε(0)) ,
λρ,1,1(κ) = λ0 + κλ(1)(ρ,1) + κ2 λ
(2)(ρ,1) +
∞∑
ν=3
a(ρ,1,1)ν κν
for ρ ∈ 1, 2 and κ ∈ Uε(0) .
In case (2)& (5) the eigenvalue λ0 of h0 splits under perturbation into one 1-cycle
(”no splitting”)
σ(h(κ)) =λ1,1,1(κ)
(κ ∈ Uε(0)) ,
λ1,1,1(κ) = λ0 + κλ(1)(1,1) + κ2 λ
(2)(1,1) +
∞∑
ν=3
a(1,1,1)ν κν (κ ∈ Uε(0)).
In case (3) the eigenvalue λ0 of h0 splits under perturbation into one 2-cycle
σ(h(κ)) =λ1,1,1(κ), λ1,1,2(κ)
(κ ∈ Dε(0)) ,
λ1,1,i(κ) = J(k1,0)1,1 (κ) + J
(k1,∞)1,1,i (κ) (i ∈ 1, 2, κ ∈ Dε(0)) ,
where
J(k1,0)1,1 (κ) = λ0 + κλ
(1)(1,1) + κ2 λ
(2)(1,1) + κ3 a
(1,1)3·2 + . . .+ κk1 a
(1,1)k1·2
,
J(k1,∞)1,1,i (κ) =
∞∑
ν=1
a(1,1,i)k1·2+ν κ
k1+ν\2 (i ∈ 1, 2)
for some k1 ≥ 2 and all κ ∈ Dε(0) .
In case (4) the eigenvalue λ0 of h0 splits under perturbation into two 1-cycles
σ(h(κ)) =λ1,1,1(κ), λ1,2,1(κ)
(κ ∈ Uε(0)) ,
λ1,l,1(κ) = λ0 + κλ(1)(1,l) + κ2 λ
(2)(1,l) +
∞∑
ν=3
a(1,l,1)ν κν
for l ∈ 1, 2 and κ ∈ Uε(0) .
Proof: See Section 2.2.4 and Section 2.2.5.
In the next proposition we list the eigenprojections and eigennilpotents for the λ0-group ofh(κ).
52 J. Rama
Proposition 2.39 Assume (A1) - (A6) with dimRanΠ0 = N = 2. Let h(κ) bethe analytic matrix family found in (2.41). Assume the results of Proposition 2.37.Depending on the factorizations (1) - (5) in Proposition 2.37 for the eigenprojectionsand eigennilpotents of h(κ) it holds:
In case (1) for ρ ∈ 1, 2 and κ ∈ Uερ(0) with some ερ > 0 small enough it holds
Πρ,1,1(κ) =
∞∑
ν=0
Π(ν)(ρ,1,1)κ
ν = Π(0)(ρ,1,1) +O(κ) (|κ| → 0)
for some Π(ν)(ρ,1,1) ∈ End(RanΠ0) and
Dρ,1,1(κ) = 0 .
In case (2)& (5) for κ ∈ Uε(0) with some ε > 0 small enough it holds
Π1,1,1(κ) = Π0 +
∞∑
ν=1
Π(ν)(1,1,1)κ
ν = Π0 +O(|κ|) (|κ| → 0)
for some Π(ν)(1,1,1) ∈ End(RanΠ0) and
D1,1,1(κ) =∞∑
ν=2
D(ν)(1,1,1)κ
ν = O(κ2) (|κ| → 0)
for some D(ν)(1,1,1) ∈ End(RanΠ0).
In case (3) for i ∈ 1, 2 and κ ∈ Dε(0) with some ε > 0 small enough it holds
Π1,1,i(κ) =∞∑
ν=−2k+3
Π(ν)(1,1,i)κ
ν/2 = O(κ−k+ 32 ) (|κ| → 0)
for some Π(ν)(1,1,i) ∈ End(RanΠ0) and
D1,1,i(κ) = 0 .
In case (4) for l ∈ 1, 2 and κ ∈ Uε(0) with some ε > 0 small enough it holds
Π1,l,1(κ) =∞∑
ν=0
Π(ν)(1,l,1)κ
ν = Π(0)(1,l,1) +O(κ) (|κ| → 0)
for some Π(ν)(1,l,1) ∈ End(RanΠ0) and
D1,l,1(κ) = 0 .
Proof: See Section 2.2.4 and Section 2.2.5.
The following proposition collects results on the dynamics e−ith(κ):
Time Evolution of Quantum Resonance States 53
Proposition 2.40 Assume (A1) - (A6) with dimRanΠ0 = 2. Let h(κ) be the analyticmatrix family defined in (2.41). Assume the results of Proposition 2.37, Proposition2.38 and Proposition 2.39. Depending on the cases (1) - (5) of factorizations in Propo-sition 2.37 one obtains the following explicit expressions for e−ith(κ) and its growthproperty:
• In case (1) for all t ∈ R and κ ∈ Uε(0) with some ε > 0 small enough it holds
e−ith(κ) =
2∑
ρ=1
e−itλρ,1,1(κ)Πρ,1,1(κ) . (2.162)
For all t ≥ 0 and 0 ≤ κ < ε for some ε > 0 small enough it holds
‖e−ith(κ)‖ ≤ 2 e−tκ2 min
ρ∈1,2|Imλ
(2)
(ρ,1)|+t·ImO(κ3)(
1 +O(κ))
(κ→ 0) . (2.163)
• In case (2) and case (5) for all t ∈ R and κ ∈ Uε(0) with some ε > 0 small enough itholds
(1 +O(κ) + tκ2O(1) + tκ3O(1)) for κ ∈ [0, ε) incase (2) and (5) ,
O(κk1+1+ 12−2k1) for κ ∈ (0, ε) in
case (3)
(2.171)
as κ→ 0.
Proof: (2.162) and (2.163) are shown in Section 2.2.4. (2.164) and (2.165) are derived inSection 2.2.4. (2.166) and (2.167) are proven in Section 2.2.5, and (2.168) and (2.169) areresults of Section 2.2.5. The estimate (2.170) is a consequence of (2.163) and (2.169). Theestimate (2.171) is a consequence of (2.165) and (2.167).
We are now prepared to formulate our main results of Part 2, given by the followingTheorem 2.41 and Corollary 2.42.
Theorem 2.41 Assume (A1) - (A6). Let h(κ) be the analytic matrix family definedin (2.41). Then:
1. There exists ε > 0 small enough such that the following operator relation on RanΠ0
2. Let dim RanΠ0 = 2. Then e−ith(κ) and ‖e−ith(κ)‖ are analyzed in detail in Proposi-tion 2.40, depending on the cases (1) - (5) of factorizations introduced in Proposition2.37. In particular there exists ε > 0 small such that for some γ > 0 and all t ≥ 0
Then inserting (2.63) into (2.182) finishes the proof of Theorem 2.41 1. (Note that (2.174)is nothing but (2.62).)
Proof of Theorem 2.41, 2.: See Proposition 2.40.
Proof of Theorem 2.41, 3.: (2.177) follows from (2.172) together with (2.173) - (2.176).
In (2.172) and (2.173) for κ small the terms O(κ2)e−ith(κ), e−ith(κ)O(κ2) andO(κ2)e−ith(κ)O(κ2) are dominated by e−ith(κ) for all t ≥ 0. So the main term is e−ith(κ).Now the question of interest is: When does in (2.172)/(2.173) the time-dependent remainderB(κ, t) dominate the main term e−ith(κ)? The answer gives the following Corollary, whichin fact is our central result:
Corollary 2.42 Assume (A1) - (A6) with dim RanΠ0 = 2 and the results of Theorem2.41. Then for times
0 ≤ t = O((− lnκ) · κ−2) (κ→ 0)
the main term e−ith(κ) dominates the time-dependent remainder B(κ, t) and thereforethe remainder R(κ, t) introduced in (2.172) and (2.173).
Proof: One has to check for which time the following statement holds: ‖e−ith(κ)‖ ≈‖B(κ, t)‖. From (2.174) obviously follows the rough estimate
‖B(κ, t)‖ ≤ cnκ2 (t ≥ 0 , κ ∈ [0, ε))
56 J. Rama
for some cn ≥ 0. Assume the estimates (2.170) and (2.171) of ‖e−ith(κ)‖. In case (1) andcase (4) set
e−κ2γt != cκ2 (2.183)
for some c ≥ 0, which gives
t = − ln c
γκ−2 +
2
γ(− lnκ)κ−2 . (2.184)
In case (3) set
e−κ2γtκ−k1+32
!= cκ2
for some c ≥ 0, which gives
t = − ln c
γκ−2 +
k1 + 12
γ(− lnκ)κ−2 . (2.185)
Note that − ln cγ κ−2 < 0 for κ > 0, since c ≥ 0 and γ > 0. But
(− lnκ)κ−2 > 0 (0 ≤ κ < 1) and limκ↓0
((− lnκ)κ−2
)= ∞ .
For 0 ≤ κ ≪ 1 the terms in (2.184) and (2.185) containing (− lnκ)κ−2 dominate the term− ln c
γ κ−2. (For example: If κ = 10−5, then κ−2 = 1010 and (− lnκ)κ−2 ∼ 1011 .)
In cases (2) and (5) one has
‖e−ith(κ)‖ ≤ e−κ2γt(1 +O(κ)) + R(κ, t) ,
where
R(κ, t) := e−κ2γt(tκ2O(1) + tκ3O(1)) .
As we have just seen e−κ2γt(1 +O(κ)) dominates B(κ, t) for 0 ≤ t = O((− lnκ) · κ−2). Butfor t = (− lnκ) · κ−2 it holds
for 0 ≤ t = O((− lnκ) · κ−2). This finishes the proof of the Corollary.
Part 3
Discussion
We shall now discuss and compare the results of Part 1 and Part 2: In Theorem 1.8,independent of dimRanΠ0 ≤ ∞, the term Π0e
−itHψ0 decays roughly exponentially up toan additive time-dependent error of order O(κ2t) +O(κ) +O(κε) for times 0 ≤ t = O(κ−2)as κ→ 0; see (1.22).
In contrast to this in the analytic setting of Theorem 2.41 the term Π0e−itH(κ)Π0 decays
roughly exponentially (cf. (2.172)), where for dimRanΠ0 = 2 the critical error terms havemultiplicative character obeying estimates which hold for all t ≥ 0; cf (2.175) and (2.176).Due to these multiplicative errors the main term e−ith(κ) takes control for times 0 ≤ t =O((− lnκ) · κ−2) (see Corollary 2.42), which is a logarithmic improvement with respect tothe results of Theorem 1.8 in the non-analytic case. This may seem weak, but acquires alot of analytic structure of the effective Hamiltonian h(κ) in (2.172).
In retrospect it seems to be impossible to get better bounds than (1.28) - (1.30) onthe (additive) errors in the non-analytic case without further assumptions on the analyticstructure of the Hamiltonian. So one can regard our estimates in Theorem 1.8 as nearoptimal.
Finally we remark that our methods and results of Part 2 also apply to some non dilationanalytic setting:
Let H(κ) = H0 + κV (κ ≥ 0 small) be a self-adjoint Hamiltonian in a complex Hilbertspace, V a symmetric perturbation (without any assumptions on dilation analyticity) andλ0 a finitely degenerate embedded eigenvalue of H0 with corresponding eigenprojection Π0,dimRanΠ0 <∞. For t ≥ 0 assume the operator relation on RanΠ0
Π0e−itH(κ)Π0 = e−ith(κ) + r(κ, t) . (3.1)
Whenever one has such a finite dimensional reduction (3.1) to RanΠ0, where the (nonself-adjoint) effective Hamiltonian h(κ) is an analytic matrix family on RanΠ0 for κ insome complex neighborhood of κ = 0, then our methods of Part 2.2 (which are basedon a Jordan decomposition of h(κ)) are applicable to this setting. Proceeding as in Part2.2 should yield results very similar to the results of Part 2.3, if the remainder sufficesr(κ, t) = O(κp) (κ→ 0) for some p > 0 large enough, uniformly in t ≥ 0, and if the followingstatements hold: h(κ) =
∑∞ν=0 h
(ν)κν for all κ in some complex neighborhood of zero andsome h(ν) ∈ End(RanΠ0) with
the eigenvalues λ(κ) of h(κ) splitting from λ0 have imaginary part Imλ(κ) < 0.
In particular our methods seem to be applicable to (a subclass of) Hamiltonians H(κ)satisfying the conditions of [JNe, Theorem 4]. This should yield the same time control asgiven in Corollary 2.42, since in [JNe, Theorem 4] the remainder (roughly comparable to theremainder B(κ, t) in our setting of Theorem 2.41) also is of order O(κ2) as κ→ 0, uniformly
57
58 J. Rama
in t ≥ 0; cf. (0.2). Unfortunately we did not notice the preprint [JNe] until an advancedstage of working out Part 2.
There seems to be no hope to get analogous results in a more generalized case corre-sponding to an eigenvalue λ0 of arbitrary finite multiplicity, where the effective Hamiltonianh(κ) does not analytically depend on κ.
Appendix A
Existence of δ(H − λ0) and P.V.(H − λ0)−1
Lemma A.1 Assume (C0) - (C4). Then δ(H−λ0) and P.V.(H−λ0)−1 exist in the sense
The proof of [MeSi, Proposition 3.2 (i)] uses (C4); see [MeSi, p.573]. The estimate (C4)might be derived from the Mourre estimate. We refer the reader to [CyFrKiSim, Chapter4] for the definition and important results of Mourre estimates.
59
Appendix B
Some Basics on Jordan Decompositions
This section collects well known results on Jordan decompositions, which are the basic toolsfor improving our results of Section 1.1 in the case of dilation analytic perturbations. Werefer the reader to, e.g., [Wu, Kapitel 22.6] or [K, Chapter I § 5 ] for more details on Jordandecomposition (or: spectral representation) of finite dimensional operators.
Let V be a finite-dimensional C-vector space, 0 < dimV =: N < ∞. Let End(V ) denotethe linear space of endomorphisms on V . Let T ∈ End(V ). Then the pairwise distincteigenvalues λ1, . . . , λs of T constitute its spectrum
σ(T ) =λj ∈ C
∣∣ j ∈ 1, . . . , s for some s ≤ N
.
Let m(λj) denote the algebraic multiplicity of the eigenvalue λj . Let d := min|λj −
λk|∣∣ j, k ∈ 1, . . . , s, j 6= k
. Then
Πj := − 1
2πi
∮
|z−λj |=r
(T − z)−1 dz (j ∈ 1, . . . , s, 0 < r < d) (B.1)
denotes the eigenprojection for T corresponding to the eigenvalue λj . The eigenprojectionshave the following properties:
Π2j = Πj and ΠjΠk = δjkΠj (j, k ∈ 1, . . . , s) ,
s∑
j=1
Πj = 1 .
The eigennilpotents for T corresponding to λj are defined by
Dj := (T − λj1 ) Πj (j ∈ 1, . . . , s) . (B.2)
It holds
Dmj = 0
(m ≥ m(λj), j ∈ 1, . . . , s
). (B.3)
Every T ∈ End(V ) has the following spectral representation (Jordan decomposition):
T =
s∑
j=1
λjΠj +Dj .
Note that T , Πj , Dj commute with each other and TΠj = ΠjT = ΠjTΠj = λjΠj + Dj
(j ∈ 1, . . . , s).
For the definitions of (semi)simplicity of eigenvalues and finite rank operators we follow [K,p.41]:
An eigenvalue λj of T is defined to be semisimple if Dj = 0. An eigenvalue λj of T is calledsimple if m(λj) = 1. Note that m(λj) = 1 implies Dj = 0.
61
62 J. Rama
T is said to be semisimple (or diagonable) if it has a spectral representation, where alleigennilpotents are zero:
T =s∑
j=1
λjΠj .
T is semisimple if and only if all its eigenvalues are semisimple. T is called simple if all theeigenvalues are simple; in this case s = N .
We close this section with a very useful spectral representation for holomorphic functions offinite dimensional operators:
Lemma B.1 [K, p.45, (5.50), (5.51)]
Let V be a finite-dimensional C-vector space, T ∈ End(V ). Let ∆ be a domain in C,σ(T ) = λj ∈ C | j ∈ 1, . . . , s for some s ≤ dimV ⊂ ∆. Let φ(·) be a holomorphicfunction in ∆. Then φ(T ) has the spectral representation
φ(T ) =
s∑
j=1
φ(λj)Πj +Dj′ , where
Dj′ :=
m(λj)−1∑
k=1
1
k!φ(k)(λj)Dj
k (m(λj) ≥ 2) .
Since Dj = 0 for m(λj) = 1, one has
Dj′ = 0 for m(λj) = 1 .
Sketch of proof: The proof of Lemma B.1 uses a decomposition of (T − z)−1 into partialfractions [K, (5.23)], which is then substituted into a definition of φ(T ) by a Dunford-Taylorintegral [K, (5.47)].
Appendix C
On Algebraic Functions
C.1 Analyzing Polynomials with Coefficients being Analytic orMeromorphic Functions on Riemann Surfaces
First we remark that all definitions in context with Riemann surfaces important for thissection are collected in Appendix D.
Let X be a Riemann surface. Let O(X)[λ] denote the ring of polynomials in one variablewith coefficients in O(X); see Definition D.4. Let M(X)[λ] denote the ring of polynomialsin one variable with coefficients in M(X); see Definition D.10. Then each P (λ) ∈ O(X)[λ](respectively ∈ M(X)[λ]) of degree n has the following representation:
P (λ) = c0λn + c1λ
n−1 + . . .+ cn
for some cj ∈ O(X) (respectively ∈ M(X)), j ∈ 0, 1, . . . , n. Polynomials with c0 = 1 arecalled normalized.
We recall that M(X) is a field; see [Fo, 1.16 Remark]. But O(X) ⊂ M(X) is an integraldomain (or: entire ring); see [Re, 4* §2, p.94 ff.].
Definition C.1 (e.g., [Bau, Anhang § 2.2])
Let K be a field and K[λ] be the ring of polynomials in one variable with coefficients in K.
1. Let P (λ), Q(λ), R(λ) ∈ K[λ] with P (λ) = Q(λ)R(λ). Then Q(λ) and R(λ) are calleddivisors of P (λ).
2. Let P (λ) ∈ K[λ] of degree n > 0.
P (λ) irreducible :⇔ There exists no divisor Q(λ) ∈ K[λ] of P (λ) with
0 < degree(Q(λ)) < n .
3. Let P (λ) ∈ K[λ].
P (λ) reducible :⇔ ¬(P (λ) irreducible
)
(We remark that this definition also applies to K being a ring.) We shall apply this forK = M(X).
63
64 J. Rama
Theorem C.2 [Fo, 8.9 Theorem]
Let X be a Riemann surface, and let
P (λ) = λn + c1 λn−1 + c2 λ
n−2 + . . .+ cn ∈ M(X)[λ]
be an irreducible (normalized) polynomial of degree n. Then there exist a Riemannsurface Y , a branched holomorphic n-sheeted covering map π : Y → X and a mero-morphic function F ∈ M(Y ) such that (π∗P )(F ) = 0. (π∗ denotes the pull back of π;see Remark C.3.) The triple 〈Y, π, F 〉 is uniquely determined (modulo a biholomor-phic mapping, see [Fo, 8.9 Theorem] for more details). 〈Y, π, F 〉 is called the algebraicfunction defined by the polynomial P .
Remark C.3 π∗ denotes the pull back of π. It is given by
π∗ : M(X) → M(Y )
f 7→ π∗f := f π ;
see [Fo, 8.2]. Thus for P , π, F and cj (j ∈ 1, . . . , n) as in Theorem C.2,
(π∗P )(F ) = Fn + (π∗c1)Fn−1 + (π∗c2)F
n−2 + . . .+ π∗cn .
Corollary C.4 Assume the conditions and notations of Theorem C.2. Furthermore,if P (λ) ∈ O(X)[λ], then F ∈ O(Y ).
Proof of Corollary C.4: Let
P (λ) = λn + c1 λn−1 + . . .+ cn ∈ O(X)[λ] .
Then in particular P (λ) ∈ M(X)[λ]. Thus Theorem C.2 yields
with F ∈ M(Y ). Since cj ∈ O(X) (j ∈ 1, . . . , n), we have by [Fo, 1.10 Remark]
π∗ : O(X) → O(Y )
cj 7→ π∗cj = cj π .
So π∗cj ∈ O(Y ), in particular it has no poles (j ∈ 1, . . . , n). Next we will show F ∈ O(Y )by counting pole orders: Assume that there exists a pole y0 ∈ Y of F , i.e., F (y0) = ∞. Andlet ordF |y0
= −k < 0; see Definition D.14. Then
F (y) =∞∑
ν=−k
αν (y − y0)
for some αν ∈ C with α−k 6= 0 and all y in some (small enough) neighborhood of y0. Thusit follows
ordFn|y0= −k · n . (C.2)
But equation (C.1) is equivalent to
Fn = −n∑
j=1
(π∗cj)Fn−j . (C.3)
Time Evolution of Quantum Resonance States 65
Therefrom it follows
ordFn|y0= ord
( n∑
j=1
(π∗cj)Fn−j
)
= ord( n−1∑
j=1
(π∗cj)Fn−j
)
= −k · (n− 1) . (C.4)
Finally, comparing (C.2) with (C.4) yields
−k · n = −k · (n− 1) ⇔ −k = 0 ,
which is a contradiction to −k < 0. Thus the meromorphic function F has no poles.
Definition C.5 [Fo, contained in 4.23]
Let X, Y be Riemann surfaces. Let π : Y → X be an n-sheeted holomorphic coveringmap. The set of all branch points of π we denote by A ⊂ Y . The set B := π(A) ⊂ X
is called the set of critical values of π.
Then the sets A and B are closed and discrete; see [Fo, 4.23]. By Theorem D.19 π takesevery value x ∈ X, counting multiplicities, n-times on Y ; see [Fo, 4.24]. That means
n =∑
y∈π−1(x)
m(π, y) (x ∈ X) ,
where m(π, y) denotes the multiplicity introduced in Definition D.17. In particular one has,if # denotes cardinality,
#(π−1(c)
)= n (c ∈ X\B) , (C.5)
#(π−1(b)
)< n (b ∈ B) . (C.6)
C.1.1 Polynomials in O(X)[λ] as polynomials in Cz − ψ(x0)[λ]
For P (λ) ∈ O(X)[λ] and π as in Theorem C.2, we are now interested in a local representationof P (λ) around some fixed point x0 in X. The following procedure gives in some sense aTaylor expansion for the holomorphic coefficients cj of P around such a fixed point x0 ∈ X:
Let x0 ∈ X be fixed. Let U(x0) ⊂ X be a small neighborhood of x0 containing no criticalvalue of π, if x0 is not a critical value of π (respectively, containing no other critical valueof π than x0, if x0 is itself a critical value of π.) Let ψ : U(x0) → ψ(U(x0)) be a charton X, where ψ(U(x0)) ⊂ C is some open subset containing ψ(x0). Since cj ∈ O(X), byDefinition D.4 the functions
cj := cj ψ−1 : ψ(U(x0) ∩X) → C (j ∈ 1, . . . , n) (C.7)
are holomorphic on ψ(U(x0) ∩X). In particular one has
P (z − ψ(x0), λ) = λn + s1(z − ψ(x0))λn−1 + . . .+ sn(z − ψ(x0)) (z ∈ ψ(U(x0))) . (C.9)
In particular P (λ) and P (λ) have the same degree. Note that from P (λ) irreducible inO(X)[λ] does not follow P (λ) irreducible in Cz − ψ(x0)[λ]. But P (λ) uniquely factorizesinto a product
P (λ) =s∏
l=1
Pl(λ)ml (C.10)
for some ml ∈ N and some s ≤ n with Pl(λ) ∈ Cz − ψ(x0)[λ] irreducible and normalized,Pl(λ) 6= Pl′(λ), l 6= l′, l, l′ ∈ 1, . . . , s. This unique factorization (C.10) is proven by thefollowing two theorems:
Theorem C.6 [F, Kapitel 6.11, p.89, Theorem]
The ring Cz of convergent Taylor series is factorial (or: a unique factorization ring).
The proof of Theorem C.6 uses the Weierstraß Preparation Theorem. We will not reproducethe proof here. We refer the reader to, e.g., [F] or [Gu, Chapter A], [GrH, Chapter 5.3, p.678ff. and Chapter 0, p.8 ff.].
Theorem C.7 [L, Chapter IV, §2, Theorem 2.3]
Let R be a factorial ring. The polynomial ring R[λ] in one variable is factorial. Itsprime elements are the primes of R and polynomials in R[λ] which are irreducible inK[λ] (if K denotes the quotient field of R) and have content 1.
If nl denotes the degree of Pl(λ), one has
s∑
l=1
ml · nl = n .
Remark C.8 We remark that in (C.10) s < n, if x0 is a critical value of π. If x0 isnot a critical value of π, then in (C.10) s ≤ n and all polynomials Pl(λ) in the uniquefactorization are of degree nl = 1.
The following Theorem gives solutions for equations like Pl(z − ψ(x0), λ) = 0.
Time Evolution of Quantum Resonance States 67
Theorem C.9 (Puiseux) [Fo, 8.14 Theorem and Remark (1)]
Let Cz denote the field of Laurent series with finite principal part. Let
P (z, λ) = λn + c1(z)λn−1 + . . .+ cn(z) ∈ Cz[λ]
be an irreducible polynomial of degree n over the field Cz. Then there exist k ∈ Z
and a Laurent series
φ(ζ) =
∞∑
ν=k
aνζν ∈ Cζ
such that
P (ζn, φ(ζ)) = 0
as an element of Cζ.Furthermore, if cj ∈ Cz (j ∈ 1, . . . , n), then φ(ζ) ∈ Cζ.
Remark C.10 (cf. [Fo, 8.14 Remarks])
1. Another formulation for the results of Theorem C.9: The equation
Cz[λ] ∋ P (z, λ) = 0
can be solved by a Puiseux-Laurent series
λ = φ( n√z) =
∞∑
ν=k
aνzν/n
with some k ∈ Z. Furthermore,
Cz[λ] ∋ P (z, λ) = 0
can be solved by a Puiseux series
λ = φ( n√z) =
∞∑
ν=0
aνzν/n . (C.11)
2. An algebraic interpretation of Theorem C.9: By the map
Cz → Cζz 7→ ζn ,
Cζ becomes an extension field of Cz of degree n. 1, ζ, ζ2, . . . , ζn−1 is abasis of Cζ over Cz. The series φ(ζ) is a root of P in this extension field. Letε be a primitive n-th root of unity, e.g. ε = e2πi/n. Then for k ∈ 0, 1, . . . , n− 1 onehas (εkζ)n = ζn and hence
P (ζn, φ(εkζ)) = 0 . (C.12)
Thus φ(εkζ) ∈ Cζ is also a root of the polynomial P . Obviously the series φ(εkζ),k ∈ 0, 1, . . . , n − 1, are distinct. Thus Cζ is a splitting field of the polynomialP (λ) ∈ Cz[λ].(For the definitions of splitting field and extension field see, e.g., [L].)
68 J. Rama
C.2 Reformulating the Classical Ideas of multi-valued AnalyticFunctions in a more Geometric Concept based on RiemannSurfaces
We shall now reformulate these results in more classical language as used, e.g., in Kato’sbook [K] (avoiding the more geometric concept of a Riemann surface, a covering map etc.).This is necessary for our purpose, since we want to quote freely results from this book.
From now on we suppose (following [K]) that the Riemann surface X is an open connectedsubset of C. Assume P (λ) ∈ O(X)[λ], π, Y and F ∈ O(Y ) as in Theorem C.2 and Corol-lary C.4.
C.2.1 Solutions of Pl(z − x0, λ) = 0 for x0 being some critical value
Let x0 := b ∈ B ⊂ X (remember that B denotes the set of critical values; see DefinitionC.5), and let
Uε(b) :=z ∈ X
∣∣ |z − b| < ε
(C.13)
for some ε > 0 small enough be a neighborhood of b containing no other critical value ofπ than b. Then, since π is an n-sheeted covering map, the preimage of Uε(b) under π is adisjoint union of open connected subsets Bl ⊂ Y ,
π−1(Uε(b)
)=
s⋃
l=1
Bl
for some s < n. (This holds, since the preimage of an open connected set is open, andevery open set decomposes into connected components.) Define the punctured, slit opendisc around b,
Dε(b) := Uε(b)\z ∈ Uε(b) | Im z = Im b, Re z ≤ Re b . (C.14)
Then π−1(Dε(b)
)∩Bl decomposes into nl connected components Cl,i:
π−1(Dε(b)
)∩Bl =
nl⋃
i=1
Cl,i . (C.15)
Next we give the
Proof of decomposition (C.15): We use the following
Theorem C.11 [Fo, 5.11 Theorem]
Let X be a Riemann surface. Let U1(0) be the open unit disk around 0 and f : X →U1(0) a proper non-constant holomorphic map which is unbranched over U1(0)\0.Then there exists a natural number k ≥ 1 and a biholomorphic mapping φ : X → U1(0)such that the diagram
X U1(0)
U1(0)
-
AAAAAU
φ
pkf
is commutative, where pk(z) := zk.
Time Evolution of Quantum Resonance States 69
Remark C.12 We remark that Theorem C.11 and its proof given in [Fo, 5.11] hold forany open disk Ur(c) with radius r > 0 and center c, as long as the proper non-constantholomorphic map f : X → Ur(c) is unbranched over Ur(c)\c.
Note that π−1(Dε(b)) ∩Bl is a Riemann surface and
πl := π π−1(Dε(b)) ∩Bl → Uε(b)
is unbranched by construction (since the branch point π−1(b) has been removed) and triviallya proper non-constant holomorphic map (since π is a covering). Thus applying TheoremC.11 and Remark C.12 yields that there exist a natural number nl ≥ 1 and a biholomorphicmapping φl : π−1(Dε(b)) ∩Bl → Uε(b) such that the diagram
π−1(Dε(b)) ∩Bl Uε(b)
Uε(b)
-
@@
@@R
φl
pnlπl
is commutative, where pnl(z) := znl . So we have πl = pnl
φl = φnl
l .
Remark C.13 Roughly speaking each Bl consists of nl sheets, which meet in π−1(b)∩Bl. The number nl coincides with the degree of Pl introduced in (C.10).
Now
πl,i := π Cl,i → Dε(b) (i ∈ 1, . . . , nl)defines an unbranched (by construction) and biholomorphic map by Theorem C.11. Then
F |Cl,i(π−1
l,i (z)) =: λl,i(z) (z ∈ Dε(b)) (C.16)
is an analytic function of z ∈ Dε(b).
Remark C.14 In classical language: Each index l ∈ 1, . . . , s describes ”a multival-ued analytic function (multi = nl)” and each λl,i(z), i ∈ 1, . . . , nl is ”a branch ofthis nl-valued analytic function”. In the classical language of [K, II § 1.2] (followingthe classical reference [Kn]), the functions λl,i(z)nl
i=1, z ∈ Dε(b), form a cycle at”exceptional” point b (since they transform one into another under analytic continu-ation around b), and the number nl of sheets of Bl is called the period of this cycle.Therefore λl,i(z)nl
i=1, z ∈ Dε(b), is also called a nl-cycle (at point b).
Remark C.15 For z ∈ Dε(b) the function λl,i(z) is a solution of Pl(z − ψ(b), λ) = 0
with ψ the identity map on X. (Pl is introduced in (C.10).) Thus by Theorem C.9,the function λl,i(z) (z ∈ Dε(b)) is a Puiseux series in
nl√z − b , (C.17)
where (C.17) should be understood as the i-th branch of nl√ · . Thus according to
(C.11) we write
λl,i(z) = φl,i(nl√z − b) :=
∞∑
ν=0
a(l,i)ν (z − b)ν/nl (C.18)
for some a(l,i)ν ∈ C and all z ∈ Dε(b).
70 J. Rama
C.2.2 Solutions of Pl(z − x0, λ) = 0 for x0 being not a critical value
Let x0 be not a critical value of π, i.e., x0 ∈ X\B. Let
Uε(x0) :=z ∈ X
∣∣ |z − x0| < ε
(C.19)
for some ε > 0 small enough be a neighborhood of x0 containing no critical value of π.Then, since π is an n-sheeted covering, the preimage of Uε(x0) under π is a disjoint unionof s open connected subsets Bl ⊂ Y ,
π−1(Uε(x0)
)=
s⋃
l=1
Bl
for some s ≤ n. Then
πl := π π−1(Uε(x0)
)∩Bl → Uε(x0)
is unbranched (since π−1(Uε(x0)
)contains no branch points of π) and trivially a non-
constant holomorphic map (since π is a covering). Then
F |Bl(π−1
l (z)) =: λl(z) (z ∈ Uε(x0)) (C.20)
is an analytic function of z ∈ Uε(x0). To stay consistent with the notation in (C.16) and(C.18) we will then write
F |Bl,1(π−1
l,1 (z)) =: λl,1(z) (z ∈ Uε(x0)) (C.21)
instead of (C.20).
Remark C.16 In analogy to Remark C.13 we then say that Bl consists of nl = 1sheet.
Then for z ∈ Uε(x0) the function λl,1(z) is a solution of Pl(z − ψ(x0)) = 0 (l ∈ 1, . . . , s)with ψ the identity map on X. (The polynomials Pl(z − ψ(x0)) ∈ Cz − ψ(x0)[λ] havebeen introduced in (C.10).) Finally, by Theorem C.9 one gets
λl,1(z) = φl,1(z − x0) :=∞∑
ν=0
a(l,1)ν (z − x0)
ν (l ∈ 1, . . . , s) (C.22)
for some a(l,1)ν ∈ C and all z ∈ Uε(x0).
Remark C.17 Following the classical language described in Remark C.14, we callλl,1(z) (z ∈ Uε(x0)) a 1-cycle at point x0.
Appendix D
Definitions in Context with Riemann Surfaces
For the sake of the reader in this section we collect some basic definitions and results incontext with Riemann surfaces. (Almost) all the following is taken from [Fo]:
Riemann surfaces are a special case of (real) two-dimensional manifolds together withsome additional complex structure which we are about to define. A (real) n-dimensionalmanifold is a Hausdorff topological space X such that every point x ∈ X has an openneighborhood which is homeomorphic to an open subset of Rn; see [Fo, p. 1 & 2].
Definition D.1 [Fo, 1.1 Definition]
Let X be a two-dimensional manifold.
1. A complex chart on X is a homeomorphism φ : U → V of an open subset U ⊂ X
onto an open subset V ⊂ C.
2. Two complex charts φi : Ui → Vi (i ∈ 1, 2) are said to be holomorphicallycompatible if the map
φ2 φ−11 : φ1(U1 ∩ U2) → φ2(U1 ∩ U2)
is biholomorphic.
3. Let I be an index set. A complex atlas on X is a system U = φi : Ui → Vi | i ∈ Iof complex charts on X which are holomorphically compatible and which coverX, i.e.,
⋃
i∈I
Ui = X .
4. Two complex atlases U and U ′ on X are called analytically equivalent if everychart on U is holomorphically compatible with every chart of U ′.
Definition D.2 [Fo, 1.3 Definition]
A complex structure on a two-dimensional manifold X is an equivalence class of ana-lytically equivalent atlases on X.
Definition D.3 [Fo, 1.4 Definition]
A Riemann surface is a pair 〈X,Σ〉, where X is a connected two-dimensional manifoldand Σ is a complex structure on X.
Conventions: (cf. [Fo, 1.4])
1. One usually writes X instead of 〈X,Σ〉 for a Riemann surface, whenever it is clearwhich complex structure is meant.
71
72 J. Rama
2. If X is a Riemann surface, then by a chart on X we always mean a complex chartbelonging to the maximal atlas of the complex structure on X.
Definition D.4 [Fo, 1.6 Definition]
Let X be a Riemann surface, Y ⊂ X an open subset. A function f : Y → C is calledholomorphic (or analytic) on Y if for every chart ψ : U → V on X (with U ⊂ X anopen subset and V ⊂ C an open subset) the function
f ψ−1 : ψ(U ∩ Y ) → C
is holomorphic in the usual sense on the open set ψ(U ∩ Y ) ⊂ C.O(Y ) denotes the set of all functions holomorphic (or analytic) on Y .
Remark D.5 [Fo, 1.7 Remark (b)]
Of course the condition in the Definition D.4 does not have to be verified for allcharts in a maximal atlas on X, just for any family of charts covering Y . Then it isautomatically fulfilled for all other charts.
Remark D.6 [Fo, 1.7 Remark (c)]
Every chart ψ : U → V on a Riemann surface X is, in particular, a complex-valuedfunction on U . (Remember that U is some open subset ofX and V some open subset ofC.) Trivially ψ is holomorphic. One also calls ψ a local coordinate (or a uniformizingparameter) and (U,ψ) a coordinate neighborhood of any point a ∈ U .
Definition D.7 [Fo, 1.9 Definition]
Let X, Y be Riemann surfaces.
1. A continuous mapping f : X → Y is called holomorphic, if for every pair of chartsψ1 : U1 → V1 on X and ψ2 : U2 → V2 on Y with f(U1) ⊂ U2, the mapping
ψ2 f ψ−11 : V1 → V2
is holomorphic in the usual sense. The set of all holomorphic mappings from X
to Y is denoted by O(X,Y ).
2. A mapping f : X → Y is called biholomorphic, if it is bijective and both f andf−1 : Y → X are holomorphic.
3. Two Riemann surfaces are called isomorphic if there exists a biholomorphic map-ping f : X → Y .
Theorem D.8 (Local Behavior of Holomorphic Mappings) [Fo, 2.1 Theroem]
Let X and Y be Riemann surfaces and f : X → Y a non-constant holomorphicmapping. Assume a ∈ X and b := f(a). Then there exist an integer k ≥ 1 and chartsφ : U → V on X and ψ : U ′ → V ′ on Y with the following properties:
1. a ∈ U , φ(a) = 0; b ∈ U ′, ψ(b) = 0.
2. f(U) ⊂ U ′.
3. The map F := ψ f φ−1 : V → V ′ is given by
F (z) = zk (z ∈ V ) .
Remark D.9 The integer k in Theorem D.8 is uniquely determined.
Time Evolution of Quantum Resonance States 73
Definition D.10 [Fo, 1.12 Definition]
Let X be a Riemann surface, Y ⊂ X an open subset. A meromorphic function on Y
is a holomorphic function f : Y ′ → C, where Y ′ ⊂ Y is an open subset, such that
1. Y \Y ′ contains only isolated points,
2. for every point p ∈ Y \Y ′ one has
limx→p
|f(x)| = ∞ .
The points of Y \Y ′ are called the poles of f . The set of all meromorphic functions onY is denoted by M(Y ).
Remark D.11 [Fo, 1.13 Remark (a)]
Let X be a Riemann surface, Y ⊂ X an open subset. Let f ∈ M(Y ) and p a poleof f . Let (U,ψ) be a coordinate neighborhood of p with ψ(p) = 0. Then f may beexpanded in a Laurent series
f =
∞∑
ν=−k
cνψν
in some neighborhood of p.
By the following Theorem, every meromorphic function from a Riemann surface X into C
can be interpreted as a holomorphic function fromX into the Riemann sphere P1 := C∪∞.(Note that P
1 is Riemann surface; see [Fo, 1.5 Remark (c)].)
Theorem D.12 [Fo, 1.15 Theorem]
Let X be a Riemann surface and f ∈ M(X). For each pole p of f define f(p) :=∞. Then f : X → P
1 is a holomorphic mapping. Conversely, if f : X → P1 is a
holomorphic mapping, then f is either identically equal ∞ or else f−1(∞) consists ofisolated points and f : X\f−1(∞) → C is a meromorphic function on X.
Next we motivate the definition for an order of a pole of a meromorphic function on aRiemann surface. This definition is not pointed out in [Fo], although it is implicitly containedin Theorem D.8:
Let X be a Riemann surface and f ∈ M(X). Let p ∈ X be a pole of f ∈ M(X). ByTheorem D.12 f : X → P1 is non-constant holomorphic. Thus by Theorem D.8.3 there existcharts φ : U → V on X and ψ : U ′ → V ′ on P1 and a natural number k ≥ 1 with
p ∈ U , φ(p) = 0 , (D.1)
∞ = f(p) ∈ U ′ , ψ(f(p)) = ψ(∞) = 0 (D.2)
and
(ψ f φ−1)(z) = zk (z ∈ V ) .
If one wants the pole not to be mapped to zero in the sense of (D.2), but to ψ(∞) 7→ ∞,one has to use the mapping
ψ(w) :=1
ψ(w)(w ∈ U ′) , (D.3)
where
ψ(∞) =1
0:= ∞ .
Then
(ψ f φ−1)(z) = z−k (z ∈ V ) .
Then the number −k < 0 is called the order of the pole p of f ∈ M(X). We summarize:
74 J. Rama
Definition D.13 Let X be a Riemann surface and f ∈ M(X) = O(X,P1). Letp ∈ X be a pole of f ∈ M(X). If k ≥ 1 is the integer found in Theorem D.8.3, then−k < 0 is called the order of the pole p. In symbols:
ord f |p = −k < 0 .
Another (equivalent) definition of the pole order is the following, motivated by Remark D.11:
Definition D.14 Let X be a Riemann surface and Y ⊂ X be an open subset. Letf ∈ M(Y ) and p ∈ Y be a pole of f . Let ψ : U → V be a chart on X and (U,ψ) acoordinate neighborhood of p with ψ(p) = 0. Then f may be expanded in a Laurentseries
f =
∞∑
−k
ανψν
in some neighborhood of p for some αν ∈ C, α−k 6= 0 for some −k < 0. The number−k is called the order of the pole p. In symbols:
ord f |p = −k < 0 .
Remark D.15 An elementary calculation shows that the Definition D.14 does notdepend on the choice of the chart.
We are now going to introduce covering maps: Covering maps are non-constant holomorphicmappings between Riemann surfaces, possibly having branch points. Therefore we define
Definition D.16 [Fo, 4.3 Definition]
Let X, Y be Riemann surfaces and f : X → Y a non-constant holomorphic map. Apoint x ∈ X is called a branch point of f , if there is no neighborhood U of x such thatf U is injective. The map f is called unbranched holomorphic, if it has no branchpoints.
Definition D.17 [Fo, 2.2 Remark]
Let X, Y be Riemann surfaces. Let f : X → Y be a non-constant holomorphic map.Let a ∈ X and b := f(a). Let k ∈ N. Then:
f has multiplicity k at point a :⇔ For every neighborhood U(a) of a there
(in symbols: m(f, a) = k) exist a neighborhood U ⊂ U(a) of a and
a neighborhood V of b = f(a) such that
#(f−1(y) ∩ U
)= k (y ∈ V \b) .
Remark D.18 [Fo, contained in 4.20]
Recall that a locally compact topological space is a Hausdorff space such that everypoint has a compact neighborhood. A continuous mapping f : X → Y between twolocally compact spaces is called proper if the preimage of every compact set is compact.A proper mapping is closed, i.e., the image of every closed set is closed.
Time Evolution of Quantum Resonance States 75
Theorem D.19 [Fo, 4.24 Theorem]
Let X and Y be Riemann surfaces and f : X → Y a proper non-constant holomorphicmap. Then there exist an n ∈ N such that f takes every value y ∈ Y , countingmultiplicities, n times on X. That means
n =∑
x∈f−1(y)
m(f, x) (y ∈ Y ) ,
where m(f, x) denotes the multiplicity introduced in Definition D.17.
Definition D.20 [Fo, 4.24 Remark]
Let X and Y be Riemann surfaces and f : X → Y a proper non-constant holomorphicmap. Let n be the integer found in Theorem D.19. Then f is called an n-sheetedholomorphic covering map.
Remark D.21 Remember that n-sheeted holomorphic covering maps are allowed tohave branch points.
Definition D.22 [Fo, Part of 17.14]
Let X and Y be Riemann surfaces and f : X → Y a proper non-constant holomorphicmap. For x ∈ X let m(f, x) be the multiplicity introduced in Definition D.17. Thenumber
b(f, x) := m(f, x) − 1
is called the branching order of f at point x.
Remark D.23 Note that b(f, x) = 0 if and only if f is unbranched at x.
Appendix E
The Ring O(X)
For the sake of the reader we include some standard facts on ideal theory of the ring ofholomorphic functions taken from [Re]. (For the definitions appearing here in this algebraiccontext we refer the reader to, e.g., [L] or [Re].)
Let X ⊂ C be an open connected subset. Let O(X) denote the ring of all functions holo-morphic in on X. (By a remark in [Re, Chapter 6 §3*.5, p.140] all the following resultsremain true, if X is a non-compact Riemann surface.)
O(X) is an integral domain (or: entire ring); see, e.g., [Re, Chapter 4* §2]. As remarkedin [Re, Chapter 4* §2.1], functions 6= 0 in O(X) with infinitely many zeros in X cannotbe written as a product of finitely many primes. Since such functions exist in every openconnected subset X ⊂ C by the General Product Theorem (see [Re, Chapter 4* §1.3, p.92]),one has: No ring O(X) is factorial. Nonetheless, O(X) has an interesting ideal-theoreticstructure:
As proved in [Re, Chapter 6 §3*.1], no ring O(X) is Noetherian. In particular, O(X) isnever a principal ideal ring.(A ring R is called Noetherian, if every ideal in R is finitely generated. A ring R is called aprincipal ideal ring, if every ideal in R is a principal ideal.)
Only the finitely generated ideals in O(X) are principal ideals, which is the statement ofthe following Theorem:
Theorem E.1 (Main Theorem of the Ideal Theory for O(X)) [Re, Chapter 6 §3*.5]
The following statements about an ideal a ⊂ O(X) are equivalent:
1. a is finitely generated.2. a is a principal ideal.3. a is closed.
(An ideal a in O(X) is called closed, if a contains the limit function of every sequence fn ⊂ a
that converges compactly in X.) The proof of this main Theorem given in [Re, p.139] usesthe Principal Ideal Theorem and therefore in fact the following Proposition an Wedderburn’sLemma:
Lemma E.2 (Wedderburn) [Re, Chapter 6 §3*.2]
Let u, v ∈ O(X) be relatively prime. Then they satisfy an equation
ϕu+ ψv = 1 with some functions ϕ, ψ ∈ O(X).
77
78 J. Rama
Proposition E.3 [Re, Chapter 6 §3*.3]
If f ∈ O(X) is a greatest common divisor (gcd) of the finitely many functions f1, . . . , fn ∈O(X), then there exist functions ϕ1, . . . , ϕn ∈ O(X) such that
f = ϕ1f1 + ϕ2f2 + . . .+ ϕnfn .
We remark that in the ring O(X), every set S 6= ∅ has a gcd; see [Re, Chapter 4* §2.1, p.95].
Every finitely generated ideal a in O(X) is a principal ideal: If a is generated byf1, . . . , fn, then a = O(X)f , where f = gcdf1, . . . , fn.
Remark E.5 [Re, contained in Chapter 6 §3*.3]
Integral domains in which any finite collection f1, . . . , fn has a gcd are called pseudo-Bezout domains. They are called Bezout domains, if this gcd is moreover a linearcombination of the f1, . . . , fn. O(X) is thus a Bezout domain.
As described in detail in [Re, Chapter 6 §3*.6], the ideal theory of the ring O(X) was notdeveloped until relatively late in the twentieth century.
Appendix F
Analyticity
In the following let X and Y be Banach spaces. C(X,Y ) will denote the set of all closedlinear operators from X to Y . By B(X,Y ) we will denote the linear space of all boundedlinear operators from X to Y . Further, ρ(·) will denote the resolvent set of an operator.
F.1 Bounded-Analyticity, Analyticity in the Generalized Sense
Definition F.1 Let X and Y be Banach spaces. Let D be an open connected subsetin C. Let
T (·) : D → B(X,Y )
κ 7→ T (κ) .
1. Let κ0 ∈ D. T (κ) is defined to be bounded-holomorphic (or: bounded-analytic)at κ0 if
∂
∂κT (κ)
∣∣∣∣κ=κ0
:= limκ→κ0
T (κ) − T (κ0)
κ− κ0
exists as a norm limit in B(X,Y ).
2. T (κ) is defined to be bounded-holomorphic (bounded-analytic) in D, if it isbounded-holomorphic (bounded-analytic) at each point κ ∈ D.
If T (κ) is not a family of bounded operators, then there is the concept of analyticity ingeneralized sense:
Definition F.2 [K, contained in VII § 1.2]
Let X, Y be Banach spaces. Assume the family of (possibly unbounded) operatorsT (κ) ∈ C(X,Y ), defined for κ in some complex neighborhood Ω of κ = 0.
T (κ) analytic in the generalized sense at κ = 0 :⇔
There exists a third Banach space Z and two families of operators U(κ) ∈ B(Z,X)and V (κ) ∈ B(Z, Y ), which are bounded-analytic at κ = 0 such that U(κ) mapsZ onto D(T (κ)) one to one and
T (κ)U(κ) = V (κ) .
for all κ ∈ Ω.
For bounded operators the definitions of bounded-analyticity and analyticity in the gener-alized sense are equivalent, as the following Lemma shows:
79
80 J. Rama
Lemma F.3 [K, contained in VII § 1.2]
Let D ⊂ C be an open connected subset. Let X, Y be Banach spaces. Let T (κ) ∈B(X,Y ), κ ∈ D. Then:
T (κ) is bounded-analytic in D, if and only if T (κ) is analytic in the generalized sensein D.
There is a relation between analyticity in the generalized sense of an operator family andthe bounded-analyticity of the corresponding resolvents:
Theorem F.4 [K, VII § 1.2, Theorem 1.3]
Let X be a Banach space. Let T (κ) ∈ C(X,X) be defined in some complex neighbor-hood Ω of κ = 0 and let z ∈ ρ(T (0)). Then:
1. T (κ) is analytic in the generalized sense at κ = 0, if and only if z ∈ ρ(T (κ))and the resolvent R(z, κ) := (T (κ) − z)−1 is bounded-analytic for |κ| sufficientlysmall.
2. If T (κ) is analytic in the generalized sense at κ = 0, then for all z ∈ ρ(T (0)) thereexists r(z) > 0 such that R(z, κ) is even bounded-analytic in both variables onthe set
〈z, κ〉
∣∣ z ∈ ρ(T (0)) , |κ| < r(z)
.
F.2 Dilation Analyticity
Let H0 := 〈L2(Rn), ‖ ·‖〉, where ‖ ·‖2 := ( · , · ) and ( · , · ) denotes the usual inner product inL2(Rn) with ( · , ψ) conjugate linear and (φ, · ) linear for fixed φ, ψ ∈ L2(Rn). Let q denotea sesquilinear form in H0 with form-domain D(q):
q[ · , · ] : D(q) ×D(q) → C ,
q[ · , ψ] conjugate linear and q[φ, · ] linear for fixed ψ, φ ∈ D(q). We abbreviate q[ψ, ψ] := q[ψ](ψ ∈ D(q)). Let H0 := −. Then H0 with operator-domain D(H0) is self-adjoint in H0.Let H+2 be the operator-domain of H0 equipped with its graph norm:
H+2 := 〈D(H0), ‖ · ‖+2〉 , ‖ψ‖2+2 :=
(ψ, (H0 + 1)2ψ
)(ψ ∈ D(H0))
Let H+1 be the form-domain of H0 with ‖ · ‖+1-norm:
H+1 := 〈D(qH0), ‖ · ‖+1〉 , ‖ψ‖2
+1 :=(ψ, (H0 + 1)ψ
)(ψ ∈ D(qH0
))
Let H−j be the completion of H0 in ‖ · ‖−j (j ∈ 1, 2), where
‖ψ‖2−2 :=
(ψ, (H0 + 1)−2ψ
), (ψ ∈ H−2)
‖ψ‖2−1 :=
(ψ, (H0 + 1)−1ψ
). (ψ ∈ H−1)
Obviously [Sim1, p.40/42]
H+2 ⊂ H+1 ⊂ H0 ⊂ H−1 ⊂ H−2 .
Remark F.5 H+1 and H+2 are reflexive Hilbert spaces.
Remark F.6 There is a pairing of H+1 and H−1 given by a (natural) sesquilinearform ( · | · ) : H+1 × H−1 → C such that (u|v) = (u, v) if v ∈ H0. But for a givenu ∈ H+1 the bounded linear functional (u| · ) = (u, · ) : H0 → C can be uniquelycontinued to a bounded linear functional (u| · ) : H−1 → C, which we will also denoteby (u, · ). H−1 can be (isometrically) identified with H+1
∗, the topological dual. This
Time Evolution of Quantum Resonance States 81
is consistent with H−1 = H0‖·‖−1
(the completion of H0 in ‖ · ‖−1), but not consistentwith
H−1 ∋ v≃7−→ ( · |v) = (v| · ) ∈ H+1
∗ .
In the following we shall identify ( · | · ) and ( · , · ). So, bounded linear operatorsT : H+1 → H−1 generate quadratic forms in H0, denoted by qT ,
qT [u] := (u, Tu) . (u ∈ H+1)
Let x := 〈x1, . . . , xn〉 ∈ Rn, let ∇ := 〈∂x1
, . . . , ∂xn〉 and
A :=1
2i(x · ∇ + ∇ · x) . (F.1)
The closure of A S(Rn) is the self-adjoint generator of the dilation group U( · ). Thisself-adjoint realization of (F.1) we denote also by A:
U(θ)±1 := e±iAθ (θ ∈ R) , (F.2)(U(θ)φ
)(x) = enθ/2φ(eθx) (φ ∈ H0 , x ∈ R
n , θ ∈ R) . (F.3)
We now recall the definitions of dilation analytic potentials, both, in operator-sense and inform-sense. The definition of dilation analytic potentials (in operator-sense) goes back toAguilar, Balslev, Combes; cf. [AgCo], [BCo]:
Definition F.7 (see, e.g., [Sim2])
Let β > 0. V ∈ B(H+2,H0) is defined to be in the class Cβ if and only if
(i) V is H0-symmetric, i.e., (u, V u) is real for all u ∈ H+2 .
(ii) V is H0-compact. (That is: V is compact as a map from H+2 to H0 .)
(iii) The family of bounded operators from H+2 to H0
V (θ) := U(θ)V U(θ)−1 (θ ∈ R) (F.4)
has an analytic continuation to a family of bounded operators from H+2 to H0
into the strip
Sβ := θ ∈ C | |Imθ| < β .(iv)
⋃
β>0
Cβ is called the family of dilation analytic potentials (in operator-sense).
The form analogue of Cβ is due to Simon [Sim2]:
Definition F.8 (see, e.g., [Sim2], [RSim4])
Let β > 0. V ∈ B(H+1,H−1) is said to be in the class Fβ if and only if
(i) V is H0-symmetric, i.e., (u, V u) is real for all u ∈ H+1 .
(ii) V is H0-form-compact. (That is: V is compact as a map from H+1 to H−1 .)
(iii) The family of bounded operators from H+1 to H−1
V (θ) := U(θ)V U(θ)−1 (θ ∈ R) (F.5)
has an analytic continuation to a family of bounded operators from H+1 to H−1
into the strip
Sβ := θ ∈ C | |Imθ| < β .(iv)
⋃
β>0
Fβ is called the family of dilation analytic potentials (in form-sense).
As Simon has proven, the class of potentials defined by operator conditions is contained inits form analogue:
82 J. Rama
Theorem F.9 [Sim2, Theorem 1]
For any β > 0, Cβ ⊂ Fβ .
Time Evolution of Quantum Resonance States 83
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