Finite dimensional Sturm Liouville vessels and their tau functions Andrey Melnikov Ben Gurion university, Israel version II 21st May 2009 Abstract Theory of Vessels was started by M. Livˇ sic in late 70’s as a part of more general theory, developed for n dimensional systems, defined by non self-adjoint commuting operators. In the Phd thesis of the author this theory is developed for the case n = 2 and there arise overdetermined time invariant systems and corresponding Vessels. A key idea of the construction is that transfer function of the system intertwines solutions of Linear Differential Equations (LDEs) with a spectral parameter. In this manner there are constructed solutions of Sturm Liouville differential equations d 2 dx 2 y(x) - q(x)y(x)= λy(x) with the spectral parameter λ and coefficient q(x), called potential. On the one hand this work can be considered as a first step toward analyzing and constructing Lax Phillips scattering theory for Sturm Liouville differential equations on a half axis (0, ∞) with singularity at 0. On the other hand, there is developed a rich and interesting theory of Vessels which has connections to the notion of τ function, arising in non linear differential equations and to the Galois differential theory for linear ODEs. The transfer function of a Vessel plays a key role in this research work. From a realization formula for the transfer function one can construct ,,tau” function τ , which is a determinant of a self-adjoint matrix function, corresponding to the SL Vessel, and can prove that there is a differential ring R* generated by τ τ ,e q , to which all the relevant objects belong. Further, using R* one can evaluate the Picard-Vessiot ring of the output LDE and so connections to the Differential Galois theory are obtained. It seems that finite dimensional SL Vessels as the most convenient environment to handle the ,,deformation theory” of Sturm Liouville differential equations. In order to precisely understand this deformation, there is studied the dependence of Vessels on spectral parameters. Contents 1 Introduction 2 2 Overdetermined time invariant 2D systems 4 2.1 Conservative Vessel [MVc] ................................. 4 2.2 Transfer function of a conservative Vessel ........................ 6 1
Transcript
Finite dimensional Sturm Liouville vessels and their tau
functions
Andrey MelnikovBen Gurion university, Israel
version II
21st May 2009
Abstract
Theory of Vessels was started by M. Livsic in late 70’s as a part of more general theory, developedfor n dimensional systems, defined by non self-adjoint commuting operators. In the Phd thesis ofthe author this theory is developed for the case n = 2 and there arise overdetermined time invariantsystems and corresponding Vessels. A key idea of the construction is that transfer function of thesystem intertwines solutions of Linear Differential Equations (LDEs) with a spectral parameter.
In this manner there are constructed solutions of Sturm Liouville differential equations d2
dx2 y(x) −q(x)y(x) = λy(x) with the spectral parameter λ and coefficient q(x), called potential. On the onehand this work can be considered as a first step toward analyzing and constructing Lax Phillipsscattering theory for Sturm Liouville differential equations on a half axis (0,∞) with singularityat 0. On the other hand, there is developed a rich and interesting theory of Vessels which hasconnections to the notion of τ function, arising in non linear differential equations and to the Galoisdifferential theory for linear ODEs.
The transfer function of a Vessel plays a key role in this research work. From a realizationformula for the transfer function one can construct ,,tau” function τ , which is a determinant of aself-adjoint matrix function, corresponding to the SL Vessel, and can prove that there is a differentialring R∗ generated by τ ′
τ, e
∫q, to which all the relevant objects belong. Further, using R∗ one can
evaluate the Picard-Vessiot ring of the output LDE and so connections to the Differential Galoistheory are obtained.
It seems that finite dimensional SL Vessels as the most convenient environment to handle the,,deformation theory” of Sturm Liouville differential equations. In order to precisely understandthis deformation, there is studied the dependence of Vessels on spectral parameters.
Sturm Liouville (SL) differential equation is [Ha] a second order differential equation with real valuedcoefficients of the form
− d
dx
(p(x)
dy(x)
dx
)+ q(x)y(x) = λw(x)y(x),
where y(x) is a function of the free variable x. Here p(x) > 0, q(x) and w(x) > 0 are specifiedand are integrable on the closed interval [a, b]. It is usually considered with separated boundaryconditions of the form
where α, β ∈ [0, π). In the Hilbert space L2([a, b], w(x)dx) there is an orthonormal basis {yi(x)} ofsolutions of this problem, corresponding to eigenvalues {λi}, where λ1 < λ2 < . . .. For example,Bessel, Legendre equations [Ha] are important example for some choices of p(x), q(x), w(x).
A special case of this equation is obtained for p(x) = w(x) = 1
d2
dx2y(x)− q(x)y(x) = λy(x), (1)
and where the parameter q(x) is usually called potential. The research of these kinds of equationsgoes back to C. Sturm [S], R. Liouville [L] in connection to dynamics, heat equation and otherphysical applications. There are many techniques developed in oder to solve this equation, whichcontributed to the study of the physics and mathematics.
The monodromy preserving deformation problem of Linear Differential Equations (LDE) wasextensively studied by Schlezinger [Shl], R. Fuchs [F] and Garnier [G] focused on Sturm Liouvilleequation particularly as the simplest non trivial LDE. See bibliography of the chapters 7,8,9 in thebook [CoLe] for more information. Scattering theory of Lax Phillips [LxPh] focused on this equationparticularly, constructing the so called spectral function for a given potential and initial conditionsy′(0)− hy(0) = 0 and scattering data. It is worth to notice the work of A. Povzner [Pov] who usedRiemann transformation to study the solutions of the PDE
∂2
∂x2u(x, y)− q(x)u(x, y) =
∂2
∂y2u(x, y)− q(y)u(x, y),
which is closely connected to the study of solutions of SL equation (1). The inverse scatteringproblem, which reconstructs the potential q(x) from the scattering data was solved by a student ofA. Povzner, V.A. Marchenko in [Mar].
2
Researches in the Differential algebra are interested in this equation (among other problems) fordetermining Liouvillian and algebraic solutions. It is the first LDE equation, where it was developedan explicit (Kovacic) algorithm [Kov] to determine existence of algebraic (over Q(t2)) solutions fora rational potential q(t2).
In the pioneer work of Moshe Livsic [Liv1] there was developed a theory of vessels, which connectsthe theory of commuting non self-adjoint operators and a theory of systems intertwining solutionsof PDEs. The theory of commuting non self-adjoint operators, developed earlier can be foundin [LKMV]. Using separation of variables in this theory one obtains that PDEs becomes LDEswith a spectral parameter [BV] and choosing coefficients [Liv2] these LDEs become Sturm Liouvilledifferential equations.
In the work [M] there is presented a theory of overdetermined 2D systems, invariant in onedirection. The transfer functions of such a system maps solutions of the input LDE with a spectralparameter λ to solutions of the output LDE with the same spectral parameter. In a special caseof such systems, LDEs are constructed from solutions of the Sturm Liouville differential equation(1). Using realization theory, developed in [M] one can construct more complicated differentialequations at the output starting from a trivial SL equation (q(t2) = 0) or more generally from SLequations with potentials, for which the solutions are ”known” (studied). This article considers finitedimensional vessels as a first step to understand the obtained potentials. As a result convergenceproblems do not arise and the tools are mostly (differentially) algebraic. Following [JMU] this can beconsidered as a ”deformation theory” of Sturm Liouville differential equations. One of the reasonsto consider it as a deformation theory is the appearance of so called tau function τ(t2), whose roleis to generate a differential ring, to which all the involved objects belong. For example the formula
for the potential at the output is q∗(t2) = −2 ∂2
∂t22log τ(t2).
The τ function is a very general concept related to non-linear differential equations. This functionwas first discovered by R. Hirota as a tool for generating many-soliton solutions (see bibliographyof [N]). For Korteveg-deVries equation
ut − 6uux + uxxx = 0
N-solitonu(x, t) = −2 ∂2
∂x2 log ∆(x, t),∆(x, t) = det
(δjk +
cjck
ηj+ηkexp[−(η3
j + η3k)t− (ηj + ηk)x]
)j,k=1,...,N
,
N-phase soliton
u(x, t) = −2∂2
∂x2log ϑ(ax + t + c),
where ϑ is the Riemann theta function. In these examples the τ -functions are ∆(x, t), ϑ(ax + t+ c)respectively. The τ function for general matrix linear differential equations with rational coefficientswas considered in [JMU]. More results concerning the τ function of the Schlesinger system can befound in [KV].
Another research area which arises in this work is differential algebra for linear differential equa-tions [PS]. It turns out that the τ function, together with a data at the input generate a differentialPicard-Vesssiot ring for the output LDE. As a result Galois differential group can be explicitly cal-culated [H]. From the point of view of differential Galois theory, there arises an interesting exampleof a finitely generated, graded differential ring, whose properties may be axiomatized and studiedin the connection to arbitrary rings (and not only over R, C as it is in this work).
This article is organized in the following way. First in section 2 overdetermined 2D systems arepresented with main theorems and realization theory for them. Then the section 3 is devoted tothe study of Sturm Liouville vessels, i.e. vessels whose input and output LDEs are derived from SLequation (1).
3
Section 3 is divided into two subsections 3.1, 3.2. At the first subsection 3.1 the so calledelementary input SL vessels are considered. Elementary means that the input LDE is the trivialone (q(t2) = 0). Tau function τ arises as a determinant of a solution of a Lyapunov equation (14),
associated to the vessel. If one denotes by R∗ the differential ring generated by τ ′
τthen it is proved
in theorem 3 that the entires of the transfer function and the potential at the output q∗(t2) are inR∗. Notice that in this simple case the solutions of the input LDE u′′(t2) = iλu(t2) are exponents,which simplifies the understanding and the proofs.
In the next subsection 3.2 it is supposed that the input differential equation is general, withsufficiently differentiable potential q(t2). In this case the input differential ring R, generated by
{e∫
q(t2)} arises and the output differential ring R∗ is now generated by two elements {e∫
q(t2), τ ′
τ}.
A generalization of theorem 3 is theorem 4 and is reproved using the notions of the input R and ofthe output R∗ differential rings. It turns out that R∗ has a structure of a graded ring (corollary12), and we can also explicitly evaluate the Picard-Vessiot ring of the output LDE, which is donein corollary 13.
At the last section 4 there is considered a dependence of all the objects on the spectral values.This is done in order to set a basis for future study of deformation theory of SL Vessels.
2 Overdetermined time invariant 2D systems
2.1 Conservative Vessel [MVc]
The notion of vessel as it appear in this article was defined by M.S. Livsic in [Liv2]. It is closelyconnected to the study of a pair of commuting non self-adjoint operators [LKMV] with compactimaginary parts and has first appeared in [Liv1]. The origins of this theory are in the fundamentalwork of M. Livsic and B. Brodskii [BL] which study the connection between non self-adjoint operatorsand meromorphic functions in the upper half plane. For each non self-adjoint operator A1 therecorresponds a naturally defined characteristic function S(λ) and vise versa. Multiplicative structureof the function S(λ) is in correspondence with invariant subspaces of the operator A1. A pair ofcommuting non self adjoint operators A1, A2 are studied via connection to their joint characteristicfunction of two variables S(λ, w) [LKMV] and there are similar results concerning invariant subspacesof both A1, A2. The notion of a Vessel arises as a collection of operators and spaces, which ”encode”the properties of A1, A2. More precisely a (conservative) vessel is
V = (A1, A2, B; σ1, σ2, γ, γ∗;H, E),
for which the following axioms holdsAj + A∗
j + BσjB∗ = 0, j = 1, 2
A2A1 −A1A2 = 0,−A2Bσ1 + A1Bσ2 + Bγ = 0,A∗
2Bσ1 −A∗1Bσ2 + Bγ∗ = 0,
γ = γ∗ + σ1B∗Bσ2 − σ2B
∗Bσ1.
Here the first axiom means that the operators are non self-adjoint, but their imaginary part may bedecomposed through an auxiliary space E . The second axiom is commutativity, the last three axiomsdetermine additional connections between factorization operators B, σ1, σ2 and some operators γ, γ∗.These results were further explored in [BV] and applied to the theory of systems. The class ofsystems, arising in this manner is very special and connects solutions of Partial Differential EquationsPDEs. The transfer function of a system (= characteristic function in Livsic’s work) maps solutionsu(t1, t2) of the input PDE
[σ2∂
∂t1− σ1
∂
∂t2+ γ]u(t1, t2) = 0
4
to solutions y(t1, t2) of the output PDE
[σ2∂
∂t1− σ1
∂
∂t2+ γ∗]y(t1, t2) = 0
with coefficients, which are constant operators. It is shown [BV] how these axioms for a two operatorvessel are derived from system theory point of view. Independence of the system transition on thepath and overdetermindness of the input/output signals is shown to be equivalent to the set of theseaxioms. These ideas has its origins in the work [Liv1].
There are also some result, considering vessels on Riemann manifolds [Ga], whose vector bundlesare Hilbert spaces.
A t1-invariant conservative vessel [MVc] is a collection of operators and spaces
V = (A1(t2), A2(t2), B(t2); σ1(t2), σ2(t2), γ(t2), γ∗(t2);H, E),
where H, E are Hilbert spaces and A1(t2), A2(t2) : H → H, B(t2) : E → H, σ1(t2), σ2(t2), γ(t2) :E → E are bounded operators, which satisfy the following axioms:
we arrive at the notion of a transfer function. Note that u(t1, t2), y(t1, t2) satisfy PDEs, butuλ(t2), yλ(t2) are solutions of LDEs with a spectral parameter λ,
λσ2(t2)uλ(t2)− σ1(t2)∂
∂t2uλ(t2) + γ(t2)uλ(t2) = 0,
λσ2(t2)yλ(t2)− σ1(t2)∂
∂t2yλ(t2) + γ∗(t2)yλ(t2) = 0.
The corresponding i/s/o system becomesxλ(t2) = (λI −A1(t2))
−1B(t2)σ1(t2)uλ(t2)d
dt2xλ(t2) = A(t2)xλ(t2) + B(t2)σ2(t2)uλ(t2)
yλ(t2) = uλ(t2)−B∗(t2)xλ(t2).
The output yλ(t2) = uλ(t2)−B∗(t2)xλ(t2) may be found from the first i/s/o equation:
yλ(t2) = S(λ, t2)uλ(t2),
6
using the transfer function
S(λ, t2) = I −B∗(t2)(λI −A1(t2))−1B(t2)σ1(t2).
Here λ is outside the spectrum of A1(t2), which is independent of t2 by (4). We emphasize herethat S(λ, t2) is a function of t2 for each λ (which is a frequency variable corresponding to t1).
Proposition 1 ([MVc]) S(λ, t2) = I −B∗(t2)(λI −A1(t2))−1B(t2)σ1(t2) has the following prop-
erties:
1. S(λ, t2) is an analytic function of λ in the neighborhood of ∞, where it satisfies:
S(∞, t2) = In×n
.
2. For all λ, S(λ, t2) is a continuous function of t2.
3. The following inequalities are satisfied:
S(λ, t2)∗σ1(t2)S(λ, t2) = σ1(t2), <λ = 0
S(λ, t2)∗σ1(t2)S(λ, t2) ≥ σ1(t2), <λ ≥ 0
for λ in the domain of analyticity of S(λ, t2).
4. For each fixed λ, multiplication by S(λ, t2) maps solutions of the input LDE with a spectralparameter λ
λσ2(t2)u− σ1(t2)du
dt2+ γ(t2)u = 0 (18)
to solutions of the output LDE with the same spectral parameter λ
λσ2(t2)y − σ1(t2)dy
dt2+ γ∗(t2)y = 0 (19)
The converse also holds (see [MVc] chapter 5 on realization problem)
Theorem 1 ([MVc]) For any functions of two variables S(λ, t2), satisfying conditions of theproposition 1, there is a conservative t1 invariant vessel whose transfer function is S(λ, t2).
Recall [CoLe] that the fourth property actually means that
S(λ, t2)Φ(λ, t2, t02) = Φ∗(λ, t2, t
02)S(λ, t02) (20)
for fundamental matrices of the corresponding equations:
λσ2(y)Φ∗(λ, y, t02)− σ1(y) ∂∂y
Φ∗(λ, y, t02) + γ∗(y)Φ∗(λ, y, t02) = 0,
Φ∗(λ, t02, t02) = I
(21)
andλσ2(y)Φ(λ, y, t02)− σ1(y) ∂
∂yΦ(λ, y, t02) + γ(y)Φ(λ, y, t02) = 0,
Φ(λ, t02, t02) = I.
(22)
As a result S(λ, t2) satisfies the following differential equation
For two gauge equivalent vessels V, V defined in (11) using the unitary operator U(t2), we shallobtain that
S(λ, t2) = I − B∗(t2)(λI − A1(t2))−1B(t2)σ1(t2) =
= I −B∗(t2)U∗(t2)(λI − U(t2)A1(t2)U
−1(t2))−1U(t2)B(t2)σ1(t2) =
= I −B∗(t2)(λI −A1(t2))−1B(t2)σ1(t2) = S(λ, t2)
But also the converse holds (see [MVc], theorem 3.5)
Theorem 2 Assume that we are given two minimal t1-invariant vessels V, V with transfer functionsS(λ, t2), S(λ, t2). Then the vessels are gauge-equivalent iff S(λ, t2) = S(λ, t2) for all points ofanalyticity.
So, if we are interested only in the transfer function, then one can bring by gauge-equivalence theoperator A1 to the simplest (up to similarity) form. We can suppose that it is a Jordan block matrixwith eigenvalues z1, . . . , zn and sizes of the corresponding blocks r1, . . . , rn. Thus if A1 is of sizeN ×N , then N = r1 + · · ·+ rn.
3 Sturm Liouville Vessels
At the first stage elementary input Sturm Liouville Vessels are considered. This case is presented inorder to prepare and discus main theorems and notions for the general case. Then arbitrary inputVessel is considered, i.e. for arbitrary q(t2).
3.1 Elementary input vessels
3.1.1 Definition of a vessel with elementary input
There exists a choice of parameters of the vessel V such that the input LDE is constructed fromsolutions of Sturm Liouville differential equation (1) with the trivial potential q(t2) = 0. Noticethat in this case the equation (1) is solved by exponents. Suppose that E = C2, i.e., a Hilbert spaceof dimension 2.
Definition 2 Sturm Liouville parameters are [Liv2]
σ1 =
[0 11 0
], σ2 =
[1 00 0
], γ =
[0 00 i
].
It easy to check that the equation (17) is satisfied. Input compatibility differential equation (18)then becomes
0 = λσ2(t2)uλ(t2)− σ1(t2)∂
∂t2uλ(t2) + γ(t2)uλ(t2) =
= λ
[1 00 0
]uλ(t2)−
[0 11 0
]∂
∂t2uλ(t2) +
[0 00 i
]uλ(t2) =
=
[λ − ∂
∂t2
− ∂∂t2
i
]uλ(t2)
and if we denote uλ(t2) =
[u1(λ, t2)u2(λ, t2)
], we shall obtain the system of equations
{λu1(λ, t2)− ∂
∂t2u2(λ, t2) = 0
− ∂∂t2
u1(λ, t2) + iu2(λ, t2) = 0
8
From the second equation u2(λ, t2) = −i ∂∂t2
u1(λ, t2) and plugging it back to the first equation,we shall obtain the trivial Sturm Liouville differential equation with the spectral parameter iλ foru1(λ, t2):
∂2
∂t22u1(λ, t2) = iλu1(λ, t2)
For the output compatibility differential (19), we take γ∗(t2) of the following form
γ∗(t2) =
[−iπ11(t2) −β(t2)
β(t2) i
]
for real valued functions π11(t2), β(t2). Consequently, for the output yλ(t2) =
[y1(λ, t2)y2(λ, t2)
], we
shall obtain that (19) is
0 = λσ2(t2)uλ(t2)− σ1(t2)∂
∂t2yλ(t2) + γ(t2)uλ(t2) =
= λ
[1 00 0
]uλ(t2)−
[0 11 0
]∂
∂t2yλ(t2) +
[−iπ11(t2) −β(t2)
β(t2) i
]uλ(t2) =
=
[λ− iπ11(t2) − ∂
∂t2− β(t2)
β(t2)− ∂∂t2
i
] [y1(λ, t2)y2(λ, t2)
]and thus the system of equations must be satisfied{
(λ− iπ11(t2))y1(λ, t2)− ( ∂∂t2
+ β(t2))y2(λ, t2) = 0,
(β(t2)− ∂∂t2
)y1(λ, t2) + iy2(λ, t2) = 0.
From the second equation y2(λ, t2) = i(β(t2)− ∂∂t2
)y1(λ, t2) and plugging it into the first equation
which means that y1(λ, t2) satisfies the Sturm Liouville differential equation with the spectral pa-rameter iλ and the potential q(t2) = (π11(t2) + β′(t2) + β2(t2)). We will see later (proposition 2)that π11 must be equal to β′(t2) + β2(t2). This condition is actually a necessary one in order tohave (16).
Definition 3 Elementary input, Sturm Liouville vessel ESL is the following collection
ESL = (A1, B(t2), X(t2); σ1 =
[0 11 0
], σ2 =
[1 00 0
],
γ =
[0 00 i
], γ∗(t2) =
[−i(β′(t2) + β2(t2)) −β(t2)
β(t2) i
];H, C2).
satisfying vessel conditions
ddt2
(B(t2)σ1) + A1B(t2)σ2 + B(t2)γ = 0, (13)
A1X(t2) + X(t2)A∗1 = B(t2)σ1B
∗(t2), (14)d
dt2X(t2) = B(t2)σ2B
∗(t2), (15)
γ∗(t2) = γ + σ1B∗(t2)X
−1(t2)B(t2)σ2 − σ2B∗(t2)X
−1(t2)B(t2)σ1. (16)
9
Remark: There arises an interval I, on which the Vessel exists, because we postulate invertibilityof the operator X(t2). For the SL case, because of positive-definiteness of σ2 it turns out that theVessel exists on a half axis.
3.1.2 τ function of an elementary input vessel
Following [MV1] (theorem 8.1) there is developed a structure of the transfer functions of a vesseland it will be applied to ESL. If SESL(λ, t2) has a realization at t02 [Br]
SESL(λ, t02) = I −B∗0 (λI −A1)
−1B0σ1,
then solving (13) with initial value B0 and (15) with X(t02) = I we obtain that
SESL(λ, t2) = I −B∗(t2)X−1(t2)(λI −A1)
−1B(t2)σ1.
Notice that since σ2 ≥ 0 and X(t02) = I > 0, we shall obtain that X(t2) > 0 for all t2 > t02 and asa result the Vessel ESL will exists at least on [t02,∞). Of course it can be extended to the left bycontinuity considerations.
Suppose that A1 = Jordan(z1, r1, . . . , zn, rn), where zi is a spectral value and ri is the size ofJordan block. Then [MV1] define companion solutions of so called adjoint output LDE
(25)and where the first vector function br1+...+ri−1(zi) is a solution of (24) with the spectral parameter−z∗i . Solving these formulas we shall obtain that
B(t2) =
b∗1b∗2...
b∗N
,
and X(t2) = [xij ] is a solution of (14). Thus the transfer function is
SESL(λ, t2) = I −B∗(t2)X−1(t2)(λI −A1)
−1B(t2)σ1 =
= I −[
b1 b2 · · · bN
]1τ[Mji](λI −A1)
−1
b∗1b∗2...
b∗N
σ1,(26)
where Mij denotes the minor i, j of the matrix X(t2) and
Definition 4 Tau function τ = τ(t2, A1) for the Jordan block matrix A1 is defined as
τ = det X(t2) = det[xij ]. (27)
10
Proposition 2 For Sturm Liouville elementary vessel the following formula for γ∗ holds
γ∗ = γ +
[−i τ ′′
τ− τ ′
ττ ′
τ0
]
Proof: From the linkage condition (16)
γ∗ = γ +1
τ
∑i,j
(−1)i+jMji
(σ1bib
∗j σ2 − σ2bib
∗j σ1
).
Suppose now that bi =
[bi1
bi2
], then the last equation becomes
γ∗ = γ +1
τ
∑i,j
(−1)i+jMji
[bi2b
∗j1 − bj1b
∗i2 −bi1b
∗j1
bi1b∗j1 0
]and it is a matter of straight calculations to show that
τ ′ =∑i,j
(−1)i+jMjibi1b∗j1, iτ ′′ =
∑i,j
(−1)i+jMji(bi2b∗j1 − bj1b
∗i2).
since from (15) x′ij = b∗i σ2bj = bi1b∗j1, and as a result x′′ij = bi2b
∗j1 − bj1b
∗i2. Notice also that
b∗i σ2bjb∗kσ2bm = b∗kσ2bjb
∗i σ2bm = b∗i σ2bmb∗kσ2bj .
3.1.3 Differential ring R∗ associated to an elementary input SL vessel
We can see that the element τ ′
τis used to construct γ∗, since τ ′′
τ= d
dt2
(τ ′
τ
)+
(τ ′
τ
)2. In the sequel
we shall use the notion of differential ring, which can be studied for example from [K]. A differentialring is a ring R with a linear operator, called derivation ∂ : R → R, satisfying the Leibnitz rule∂(ab) = (∂a)b + a∂b and such that ∂R ⊆ R. Notice [K] that intersection of two differential ringsis again a differential ring, thus we may do the following definition. The ring R is called generatedby the set {r1, . . . , rn} (n may be ∞) if R is the minimal (in inclusion) differential ring containing{r1, . . . , rn}.
Definition 5 The differential ring R∗ is generated by { τ ′
τ, 1}.
We define a space T , which plays an important role in analyzing R∗. Recall thatA1 = Jordan(z1, r1, . . . , zn, rn) and N = r1 + · · ·+ rn,
Definition 6T = span b∗i1Ei1j1bj1 . . . b∗iN
EiN jN bjN ,
where E with different indexes are 2× 2 matrices over C and in the construction of T we permit totake elements b∗i1Ei1j1bj1 subject to the following restrictions
1. for each chain of solutions of length ri, exactly ri elements bj will appear. The same is truefor b∗j ,
2. suppose that bj1 , . . . , bjriappear and are ordered by indexes, i.e. j1 ≤ · · · ≤ jti . Then jk ≤ k
for each 1 ≤ k ≤ ri. The same is true for b∗j .
We shall call each function b∗i1Ei1j1bj1 . . . b∗iNEiN jN bjN , satisfying these two condition as a basic
element.
11
Notice that the space T is constructed from basic elements, which are multiplications of exactlyN elements of the form b∗i Eijibji . If one consider only the indexes j1 ≤ · · · ≤ jri then the secondcondition means that point wise (in each coordinate) the ri tuple (j1, . . . , jri) is less or equal to(1, . . . , ri). These two restrictions on appearance of each bi, b
∗i comes from the properties of τ ,
which are explored in the next lemma.
Lemma 1 1. T ′ ⊆ T ,
2. dim T < ∞ and spann∈N(τ (n)) = T ,
3. 1τT ⊆ R∗.
Proof:
1. Using Leibnitz rule for the derivative of multiplication of functions, it is enough to prove thatthe derivative of b∗i Eijibji is a linear combination of elements of the same form. Indeed, if bi isa companion solution corresponding to the spectral parameter z (which is usually of the form−z∗i ), then
b′i = σ−11 (σ2z + γ)bi + σ2bi−1.
Suppose also that bji is a companion solution corresponding to the spectral parameter w. Then
ddt2
b∗i Eijibji = b∗i (σ2z∗ − γ)σ−1
1 Eijibji + b∗i−1σ2σ−11 Eijibji+
+b∗i Eijiσ−11 (σ2w + γ)bji + b∗i Eijiσ
−11 σ2bji−1 =
= b∗i [(σ2z∗ − γ)σ−1
1 Eiji + Eijiσ−11 (σ2w + γ)]bji + b∗i−1σ2σ
−11 Eijibji + b∗i Eijiσ
−11 σ2bji−1.
(28)and again we obtain elements of the same form. In order to see that we stay at the space T ,notice that differentiating b∗i Eijibji we obtain an element of the same form
b∗i [(σ2z∗ − γ)σ−1
1 Eiji + Eijiσ−11 (σ2w + γ)]bji
and two elements with smaller indexes
b∗i−1σ2σ−11 Eijibji , b∗i Eijiσ
−11 σ2bji−1.
But if bi or bj are initial members at a companion chain of solutions, then these two elementsdoes not exist. Thus we conclude that if we are given a basic element b∗i1Ei1j1bj1 . . . b∗iN
EiN jN bjN ,then its derivative will substitute each b∗i Eijbj with b∗kEklbl, where bi, bk and bj , bl are mem-bers of the same chain, i.e. the first condition 6.1 will hold. In order to see that the secondcondition 6.2 holds, notice that if there appear bj1 , . . . , bjri
satisfying this condition, then thederivative of b∗i1Ei1j1bj1 . . . b∗iN
EiN jN bjN will only decrease indexes and as a result preservethis condition.
2. Let us consider solutions of the adjoint output companion LDE with the spectral parameterλ (25), j = 1, . . . , ri − 1
[σ1d
dt2− µσ2 − γ]bj+1 = σ2bj
The first element in the chain, b1 is a solution of the adjoint output LDE (24). Then
b1 =
[A1e
kt2 + B1e−kt2
−i(A1ekt2 −B1e
−kt2)
], k =
√iλ
and further companion solution bi are of the form
bi =
[pi,1(t2)e
kt2 + qi,1(t2)e−kt2
pi,2(t2)ekt2 + qi,2(t2)e
−kt2
].
12
where pi,1(t2), qi,1(t2), pi,2(t2), qi,2(t2) are polynomials in t2 of exact degree i. Consequently,
each entryb∗i σ1bj
z∗i +zjof X(t2) is a linear combination of exponents multiplied by polynomials of
prescribed degrees. Moreover, opening determinant of X(t2) in order to evaluate τ(t2), weshall obtain all possible multiplications of these exponents and corresponding polynomials
where P (t2) is the multiplication of all polynomials. Notice that some elements, after openingdeterminant can be the same, depending on the values of the spectral values. Any basicelement of T is by definition a multiplication of these exponents and polynomials. Moreover,by the construction, τ consists all possible exponents. Its derivatives are of the same form, i.e.linear combinations of (29), since derivatives of these elements give their linear combinations.Since there is a finite number of such elements dim T < ∞.
Consider now τ . Each ri tuple (bj1 , . . . , bjri) is constructed from the exponents, multiplied
by polynomials of degrees j1, . . . , jri . Consequently, after their multiplication, a polynomialof the maximal degree from these elements will be obtained for (1, . . . , ri). This is exactly thecase in the definition of τ , i.e. we obtain maximally possible polynomials after opening thedeterminant. For any other ri tuple we shall obtain a polynomial of a smaller degree. Thus τis a sum of all possible exponents, multiplied by maximally (in degree) possible polynomials.Using generalized Van Dermond determinant all its derivatives are independent and span thewhole space, generated by exponents and powers of t2 with prescribed degrees, which is T .
3. In order to prove the third part, notice that 1 = ττ
is in R∗ by definition. We will show thatτ(n)
τ∈ R∗ by induction on n ≥ 1. For n = 1 it is true by the definition, and generally,
τ (n+1)
τ=
d
dt2
(τ (n)
τ
)+
τ (n)
τ
τ ′
τ,
which finishes the proof.
Theorem 3 1. T is finite dimensional, τ ∈ T satisfies a linear differential equation with constantcoefficients of finite order,
2. The entries of γ∗ and of the transfer function SESL(λ, t2) of ESL are in R∗,
Proof:
1. Since T is a linear combination of exponents multiplied by polynomials, its derivatives will beof the same form and as a result it will be a finite dimensional space. τ ∈ T by definitionand by lemma 1 all its derivatives belong to T , which is finite dimensional. Consequently, τsatisfies a linear differential equation with constant coefficients of finite order.
2. Let us use the formula (26):
SESL(λ, t2) = I −
b1
b2
...br1+···+rn
1
τ[Mji](λI −A1)
−1 [b∗1 b∗2 · · · b∗r1+···+rn
]σ1.
Notice first that
X−1(t2)(λI −A1)−1 =
1
τ[Mji](λI −A1)
−1 = [Kij ]
13
where Kij are linear combinations ofMji
τ. Then
S(λ, t2) = I −
b1
b2
...br1+···+rn
X−1(t2)1τ[Mji](λI −A1)
−1[
b∗1 b∗2 · · · b∗r1+···+rn
]σ1 =
= I −∑
ij bib∗j Kijσ1.
Since bib∗j Kij has entries in 1
τT , we obtain by lemma 1 that these entries are in R∗.
The fact γ∗ ∈ R∗ follows proposition 2 and lemma 1.
Let us finish this discussion with some properties of the differential ring R∗
Corollary 7 R∗ is a finitely generated, graded differential ring
R∗ = C +Tτ
+T 2
τ2+ · · · .
for which the derivative respects the following rule
d
dt2(T i
τ i) ⊆ T i
τ i+T i+1
τ i+1,
Proof: By the definition R∗ is generated by τ ′
τ. The grading on R∗ defined above follows since
Tτ⊆ R∗ and there are no elements of the ring with fewer multiplications of b∗j Eijbi. Linear
combinations and multiplications obviously respect the grading, since T i
τiT j
τj ⊆ T i+j
τi+j . All the otherelements of the ring are obtained by multiplication. Let us evaluate the differentiation of an elementof the grading
d
dt2(T i
τ i) =
ddt2T i
τ− i
T i
τ i+1τ ′ ⊆ T i
τ i+T i+1
τ i+1,
where ddt2T i ⊆ T i holds by lemma 1.
One can also calculate the Picard Vessiot differential ring of the output LDE. This ring is bydefinition the minimal differential ring, containing the entries of the fundamental matrix. See thereferences [H, PS] for more material on this subject.
Corollary 8 Let λ 6∈ {z1, . . . , zn} be a parameter. The Piccard-Vessiot ring of the output LDE (19)
Y ′ = σ−11 (σ2λ + γ∗(t2))Y (t2)
is generated by { τ ′
τ, ekt2 , e−kt2}, where k =
√iλ.
Proof: The fundamental matrix of the input LDE (18) can be taken as follows
Φ(λ, t2) =
[ekt2 e−kt2
−ikekt2 ike−kt2
].
Moreover from (20) we obtain that the fundamental matrix of the output LDE (19) is
Φ∗(t2) = S(λ, t2)Φ(λ, t2)S−1(λ, t02).
Since the entries of S(λ, t2) are generated by τ ′
τand the entries of Φ(λ, t2) are combinations of
{ekt2 , e−kt2}, we obtain the desired result.
14
3.1.4 Inverse problem for elementary input vessels
Let us consider now the inverse problem of the Sturm Liouville equation. Suppose that we are givenγ∗ and want to construct a vessel, for which this γ∗ defines the output LDE (19). Notice that weimmediately find the function τ from the 1, 2 element by solving the differential equation
τ ′
τ= (γ∗)12.
Once we have τ function, following formula (27) it is necessary to find the spectral values zi and thesize of the corresponding Jordan block ri in order to construct this τ . On the other hand, theorem3 gives us a necessary condition for such a τ to satisfy. It is a differential equation with constantcoefficients. Suppose that can find this differential equation and rewrite it as a polynomial in d
dt2.
Substituting further ddt2
by x, we shall obtain a polynomial p(x) in the variable x. Notice thatsince τ(t2) is the determinant of a positive definite matrix X(t2), the polynomial p(x) has to bedecomposable as
p(x) = p1(x)p∗1(x),
where p1(x) is a general complex valued polynomial. p1(x) can decomposed into a product ofn1 + · · ·+ nP elementary factors
p1(x) =
P∏i=1
(x− ki)ni , ki ∈ C, ni ∈ N. (30)
Thus we have to find first complex values in the right half plane zi such that
λi − λj = kij
for some reindexing of the elements ki. This means that a necessary condition for this to hold isthat P = n2, n ∈ N. We obtain in this manner a combinatorial problem
Problem 1 Given n2 complex numbers ki, find n numbers zi in the upper half plane such thatλi − λj = kij for some reindexing of the elements ki.
Second step is to handle the powers ni at the decomposition (30). Each zi comes with power ri,which determines the size of the Jordan block in the final A1. Thus we have to solve the followingproblem
Problem 2 Find natural numbers ri (6= 0) such that ri + rj = nij, where the indexing of ni comesfrom the solution of the previous problem.
3.2 General Sturm Liouville Vessels
In this section we want to consider an arbitrary input Sturm Liouville Vessel. In order to obtain aSturm Liouville equation at the input we will take
γ =
[−i η′′
η− η′
ηη′
ηi
]
15
for some sufficiently differentiable function η = η(t2). Then entries of the input u(t2) =
[u1(λ, t2)u2(λ, t2)
]will satisfy
∂2
∂t22u1(λ, t2)−
d2
dt22
(log η
)u1(λ, t2) = iλu1(λ, t2), (31)
u2(λ, t2) = i(η − ∂
∂t2)u1(λ, t2), (32)
which means that u1(λ, t2) satisfies the Sturm Liouville differential equation (1) with potentiald2
dt22
(log η
)and the spectral parameter iλ.
Definition 9 A general Sturm Liouville vessel is a collection
GSL = (A1, B(t2), X(t2); σ1, σ2, γ =
[−i η′′
η− η′
ηη′
ηi
], γ∗(t2);H, C2)
satisfying the vessel condition (13), (14), (15), (16).
If A1 = Jordan(z1, r1, . . . , zn, rn) with zi’s spectral values and ri’s the corresponding sizes of Jordanblocks, then defining companion solutions bi of (25) we obtain solving (13)
B(t2) =[
b∗1 b∗2 · · · b∗r1+···+rn
].
X(t2) = [xij ] is a solution of the Lyapunov equation (14) and satisfies (15). We will use the samedefinition as in the elementary input case τ = det X(t2) and then the same prove for the proposition
2 except for the new formula x′′ij = −i(bi2b∗j1 − bj1b
∗i2) + 2 η′
ηb∗i,1bj,1 gives
γ∗ =
[−i η′′
η− η′
ηη′
ηi
]+
[−i
(τ ′′
τ+ 2 η′
ητ ′
τ
)− τ ′
ττ ′
τ0
]. (33)
From here we can see that the differential ring R∗ must include the element η′
η. In the sequel, we
will also use the ring, generated by η′
ηitself, so we make the following
Definition 10 The input differential ring R is generated by { η′
η, 1}. The output differential ring
R∗ is generated by { τ ′
τ, η′
η, 1}.
This definition is a generalization of the definition 5, because in the elementary input case we obtain
that the input differential ring is generated by η′
η= 0 and 1, i.e is trivial and as a result the output
differential ring is generated only by τ ′
τand 1.
Let us define an analogue of the space T appearing in definition 6 in the following way
Definition 11TG = span b∗i1Ei1j1bj1 . . . b∗iN
EiN jN bjN R,
where E with different indexes are 2× 2 matrices over C and in the construction of T we permit totake basic elements b∗i1Ei1j1bj1 subject to the following restrictions
1. for each chain of solutions of length ri, exactly ri elements bj will appear. The same is truefor b∗j ,
16
2. suppose that bj1 , . . . , bjriappear and are ordered by indexes, i.e. j1 ≤ · · · ≤ jti . Then jk ≤ k
for each 1 ≤ k ≤ ri.
Remark: Using definition 6 of the space T , we obtain that TG = TR.In the contrary to theorem 3 this space will not usually be finite dimensional, because the input
differential ring R is generally infinite dimensional. An analogue of lemma 1 is as follows
Lemma 2 1. T ′G ⊆ TG,
2. alg-span(τ (n)R) = TG, where alg-span stands for the algebraic span.
3. 1τTG ⊆ R∗.
Proof:
1. From the definition it follows that TG = TR and using Leibnitz rule,
T ′G ⊆ T ′R + TR′.
Consequently, it is enough to show that T ′ ⊆ TR, since then
T ′R + TR′ ⊆ TRR + TR ⊆ TR ⊆ TG
as desired. In order to see that T ′ ⊆ TR, we use Leibnitz rule and evaluate the derivatives ofb∗i Eij1bji using the formula appearing in lemma 1
ddt2
b∗i Eijibji = b∗i [(σ2z∗ − γ)σ−1
1 Eiji + Eijiσ−11 (σ2w + γ)]bji + b∗i−1σ2σ
−11 Eijibji + b∗i Eijiσ
−11 σ2bji−1 =
= b∗i [
[η′
ηz∗ + i η′′
η
−i − η′
η
]Eiji + Eiji
[η′
ηi
w − i η′′
η− η′
η
]]bji + b∗i−1σ2σ
−11 Eijibji + b∗i Eijiσ
−11 σ2bji−1 =
= b∗i [
[1 00 −1
]Eiji + Eiji
[1 00 −1
]]bji
η′
η+ b∗i [
[0 z∗
−i 0
]Eiji + Eiji
[0 iw 0
]]bji+
+b∗i [
[0 i0 0
]Eiji + Eiji
[0 0i 0
]]bji
η′′
η+ b∗i−1σ2σ
−11 Eijibji + b∗i Eijiσ
−11 σ2bji−1,
which means that we obtain elements of the form b∗i Eij1bjiR. In order to see that we stay atthe space TG, notice that if bi or bji are initial members at a companion chain of solutions,then the two elements with smaller indexes
b∗i−1σ2σ−11 Eijibji , b∗i Eijiσ
−11 σ2bji−1
does not appear, which explains why the restriction on TG in definition 11 holds.
2. Suppose that there is t02 for which q(t02) = 0. Then there exists an ε neighborhood of t2 inwhich the solutions (and its derivatives) of the adjoint output LDE (24) with the spectralparameter −z∗i are close to solutions (and its derivatives) of the trivial SL equation with thesame spectral parameter. Since we have proved in lemma 1 that for the trivial case derivativesof τ generate the whole space, i.e. if we take less then dim T derivatives, they are independent,the same will hold for the modified τ function, and as a result its derivatives span the wholespace TG.
3. The same as in lemma 1.
Generalization of theorem 3 is as follows
Theorem 4 1. τ satisfies a linear differential equation of finite order over R,
2. The entries of γ∗ and of the transfer function SGSL(λ, t2) of the vessel GSL are in R∗.
17
Proof:
1. We have seen in lemma 2 that alg-span(τ (n)R) = TG. On the other hand, the space T isfinite dimensional. Thus considering the ring R) as coefficients, we shall obtain that a finitenumber of derivatives of τ are linearly dependent. Notice that there is no need to invert theseelements and we obtain the desired result.
2. Use the proof of 3.2 using TG instead of T and the formula (33) for γ∗.
The following corollaries are proved very similar to the elementary input case
Corollary 12 R∗ is a finitely generated, graded differential ring:
R∗ = C +TG
τ+T 2
G
τ2+ · · ·
for which the derivative respects the following rule
d
dt2(T i
G
τ i) ⊆ T i
G
τ i+T i+1
G
τ i+1,
Corollary 13 The Piccard-Vessiot ring of the output LDE (19) (for λ 6∈ {z1, . . . , zn})
d
dt2Y = σ−1
1 (σ2λ + γ∗(t2))Y (t2)
is generated by τ ′
τand the entries of Φ(λ, t2).
4 Dependence on the spectral values
We have defined τ function, as a function of A1, which means that it is actually a function ofz1, . . . , zn and of the sizes r1, . . . , rn of Jordan blocks. In many related problems (such as Schlesingersystems [JMU]) where τ function arises, it usually satisfies a PDE depending on these spectral values.In our case, from theorem 4 it follows that τ satisfies a linear differential equation (with respect tot2) with coefficients depending on z1, . . . , zn, r1, . . . , rn. Let us also find a defining set of LDEs andPDEs for S(λ, t2). First, we have the LDE (23) for S(λ, t2). In order to find others, we can use[MV1] the realization theorem 8.3 from where it follows that
S(λ, t2) = I −n∑
i=1
[ci,1 ci,2 . . . ci,ri
](λI − Jordan(zi))
−1
b∗i,1b∗i,2...
b∗i,ri
σ1,
where bi,k is a chain corresponding to the adjoint output LDE with the spectral parameter −z∗i (24)
[σ1d
dt2+ z∗i σ2 − γ]bi,1 = 0, [σ1
d
dt2+ z∗i σ2 − γ]bi,k+1 = σ2bi,k, 1 < k ≤ ri
and ci,k is a chain corresponding to the output LDE with the spectral parameter zi (19)
[σ1d
dt2− ziσ2 − γ∗]ci,1 = 0, [σ1
d
dt2− ziσ2 − γ∗]ci,k+1 = σ2ci,k, 1 < k ≤ ri
18
From this realization it follows that S(λ, t2) is constructed from n functions, each depending on zi
separately
S(λ, t2) = I−n∑
i=1
Si(λ, t2, zi), Si(λ, t2, zi) =[
ci,1 ci,2 . . . ci,ri
](λI−Jordan(zi))
−1
b∗i,1b∗i,2...
b∗i,ri
σ1.
It is also immediate that each one of this functions Si(λ, t2, zi) is rational in λ. Notice further thatone can choose bi,k+1 = ∂
Finally, it is important to notice that the parameters ri determine the maximal powers of λ appearingat each Si(λ, t,zi). Thus we have proved the following theorem
Theorem 5 Suppose that we are given a finite dimensional SL vessel
GSL(A1, B(t2), X(t2); σ1, σ2, γ =
[−i η′′
η− η′
ηη′
ηi
], γ∗(t2);H, C2)
for which A1 = Jordan(z1, r1, . . . , zn, rn). Then its transfer function S(λ, t2)
1. satisfies the differential equation equation (23)
∂
∂t2S(λ, t2) = σ−1
1 (σ2λ + γ∗)S(λ, t2)− S(λ, t2)σ−11 (σ2λ + γ),
2. has a decomposition S(λ, t2) = I −∑n
i=1 Si(λ, t2, zi), where each function Si(λ, t2, zi) is arational in λ function with denumerator of degree ri,
3. Si(λ, t2, zi) satisfies PDE (34)
∂∂t2
∂∂zi
Si(λ, t2, zi) = σ−11 (σ2zi + γ∗)[
∂∂zi
Si(λ, t2, zi) + ∂∂λ
Si(λ, t2, zi)]−−[ ∂
∂ziSi(λ, t2, zi) + ∂
∂λSi(λ, t2, zi)](σ2zi + γ)+
+σ−11 σ2Si(λ, t2, zi)− Si(λ, t2, zi)σ2.
In the future, it would be interesting to start from a function satisfying the condition of this theoremand try to construct the corresponding Vessel. More generally, it could be interesting to characterizeVessels with possibly continuous spectrum, whose transfer functions satisfy the last theorem exceptfor the rationality in λ.
5 Conclusions and remarks
It is possible to generalize all the formulas appearing in this article to solutions of differentialequations of greater order. For example, defining
σ1 =
0 0 10 1 01 0 0
, σ2 =
1 0 00 0 00 0 0
, γ =
iπ β α−β∗ 0 1−α −1 0
for real valued α(t2), π(t2), one obtains that for the input function uλ(t2) =
u1(λ, t2)u2(λ, t2)u3(λ, t2)
the first
entry satisfies
−u′′′1 + u′1(β∗ + β − 2c′ + c2) + u1((β
∗)′ − c′′ − iα + cc′ + cβ − cβ∗) = λu1
which is a general linear differential equation of order 3
u′′′ + q1u′ + q2u = λu.
For the output yλ(t2) =
y1(λ, t2)y2(λ, t2)y3(λ, t2)
, the first entry satisfies
y′′′1 + q1∗y′1 + q2∗y1 = λy1
21
and there are relations between q1∗, q2∗ and q1, q2.More generally, defining σ1 as anti diagonal, σ1 = E11 and γ of the same form as for n=3, we
can study differential equations of order n.We can also conclude that in the Sturm Liouville case the role of τ -function is a ”generating
element” of a universe (a generator of the output differential ring R∗ together with η′
ηcorresponding
to the input), where all the relevant objects (γ∗, transfer function, the potential q∗(t2)) live.
6 Acknowledge
I would like to thank Victor Vinnikov and Daniel Alpay for very helpful discussions and remarks. Iwould also like to thank Victor Katsnelson for helping me with material on tau functions.
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