Supersymmetric quantum mechanics and itsapplications

C.V. Sukumar

Wadham College, University of Oxford, Oxford OX1 3PN, England

Abstract. The Hamiltonian in Supersymmetric Quantum Mechanics is defined in terms of chargesthat obey the same algebra as that of the generators of supersymmetry in field theory. The conse-quences of this symmetry for the spectra of the component parts that constitute the supersymmetricsystem are explored. The implications of supersymmetry for the solutions of the Schrödinger equa-tion, the Dirac equation, the inverse scattering theory and the multi-soliton solutions of the KdVequation are examined. Applications to scattering problems in Nuclear Physics with specific refer-ence to singular potentials which arise from considerations of supersymmetry will be discussed.

1. SUPERSYMMETRIC QUANTUM MECHANICS OFONE-DIMENSIONAL SYSTEMS

It is shown that every one-dimensional quantum mechanical HamiltonianH can havea partnerH such thatH and H taken together may be viewed as the components ofa supersymmetric HamiltonianH. The term ‘supersymmetric Hamiltonian’ is taken tomean a Hamiltonian defined in terms of charges that obey the same algebra as that ofthe generators of supersymmetry in field theory. The consequences of this symmetry forthe spectra ofH andH are explored. It is shown how the supersymmetric pairing maybe used to eliminate the ground state ofH, or add a state below the ground state ofH ormaintain the spectrum ofH. It is also explicitly demonstrated that the supersymmetricpairing may be used to generate a class of anharmonic potentials with exactly specifiedspectra.

1.1. Introduction

In field theory, supersymmetry is a symmetry that generates transformations betweenbosons and fermions. Unlike the generators of other symmetries whose algebra involvescommutators, the generators of the supersymmetric transformations are spinor chargeswhose algebra involves anticommutators. Supersymmetry has raised the possibility ofproviding a framework for a unified description of bosons and fermions which arecombined in the same supersymmetric multiplet [1]. Supersymmetric field theories maybe constructed by defining a superfield in a superspace, a space consisting of the usualspacetime and in addition the anticommuting spinors of Grassmann [2]. The superfieldφis a function of the spacetime coordinatesx and alsoθ andθ whereθ is an odd memberof the Grassmann algebra andθ is its conjugate. The supersymmetric transformation

may be viewed as a Grassmann even translation in this superspace. The generators ofthis transformation are the supercharges.

There is a well defined procedure for starting from a field theory to construct a singleparticle Quantum Mechanics andvice versa. For example Quantum Mechanics in onedimension is defined by the HamiltonianH = p2 +V(x) and the commutation relation[x, p] = ih. The corresponding field theory starts by defining a space-time and the fieldφ(x, t) is defined in this space-time by the LagrangianL = [(∂tφ)2−V(φ)] and the actionS=

∫Ldt. It is well known thatd = 1 quantum mechanics is formally equivalent to the

d = 1 quantum field theory with the identificationx→ φ , p→ ∂tφ and canonical quanti-zation of the fieldφ leads to the usual commutation relations betweenx andp. Similarly,by constructing a Lagrangian invariant under the supersymmetric transformation,i.e.bygeneralizing thed = 1 field to the superfield defined in superspace and integrating outthe Grassmann coordinates associated with the superspace, a Lagrangian expressed interms of the component fields of the superfield may be obtained. Canonical quantizationthen leads to a Hamiltonian for Supersymmetric Quantum Mechanics. Witten [3] wasthe first to construct a simple example of a supersymmetric system corresponding toa spin−1

2 particle moving in one dimension. Witten also defined the algebra that mustbe satisfied by the charge operators in terms of which the supersymmetric Hamiltonianmay be expressed. These algebraic relations that Witten first formulated have now be-come the defining relations of Supersymmetric Quantum Mechanics or SUSYQM inabbreviation.

The word ‘supersymmetry’ was originally used to denote a symmetry built into cer-tain field theories that permits transformations between component fields whose intrin-sic spins differ by1

2h. However, by extracting a single particle Quantum Mechanicsfrom the field theory by integration of the Grassmann variables all reference to spinis lost. What remains is an underlying symmetry of the Schrödinger differential equa-tions for two related Hamiltonians. In fact, already in the 19th century a symmetry ofsecond-order differential equations had been identified by Darboux [4]. The Darbouxtransformation relates the solutions of a pair of closely linked second order differentialequations (Schrödinger [5], Infeld and Hull [6]). Throughout these lectures the term ‘su-persymmetric system’ will be used to describe systems governed by an underlying alge-bra which is identical to, or derivable from, the algebra of supersymmetry in field theoryeven if the systems under consideration have nothing to do with bosons and fermions asthey are commonly understood. This algebra is the algebra explicitly defined by Witten.

The study of the relationship between spectra, conservation laws and the existence ofoperators that commute with the Hamiltonian has a long history. It is well known that theconservation of energy, linear momentum and angular momentum arise when space-timeis homogeneous and isotropic which in turn lead to invariance of the Hamiltonian undertime translation, spatial translation and rotation. The invariance of the supersymmetricHamiltonian under translations in superspace is related to the existence of superchargesthat commute with the SUSY Hamiltonian and leads to definite relations between thespectra of the bosonic and fermionic sectors. Just as in field theory supersymmetryleads to specific relations between the component sectors of a supermultiplet, so alsoin SUSYQM the existence of a generating operator that commutes with the Hamiltonianleads to certain specific relations between the spectra and the eigenfunctions of thecomponent parts of the supersymmetric Hamiltonian. The links between the solutions

of two differential equations connected by the Darboux transformation are identical tothose arising from considerations of supersymmetry.

Witten’s seminal idea has now been developed into the subject of SupersymmetricQuantum Mechanics: the study of quantum mechanical systems governed by an algebraidentical to that of supersymmetry in field theory. A number of people have playedan important role in the development of the subject. It will not be possible in theselectures to do full justice to all the people who have contributed to this subject. I wouldlike to keep a chronological order of how the ideas have developed and refer to thepapers that act as markers in this progression. Witten’s 1981 paper was followed by otherexamples of spin systems in magnetic fields and other such special systems that exhibitedan underlying supersymmetry ([7]-[14]). Bernstein and Brown [15] showed that byexploiting the degeneracy between the ‘bosonic’ and ‘fermionic’ sectors of certainone-dimensional Hamiltonians, the properties of the first excited state of the ‘bosonic’component may be inferred from a knowledge of the ground state of the ‘fermionic’component. It was then shown by Andrianov, Borisov and Ioffe [16] and Sukumar[17, 18] that all one-dimensional systems can have supersymmetric partners. Andrianov,Borisov and Ioffe [19] also showed that a simple extension of supersymmetric quantummechanics to arbitrary dimensions is possible.

The plan of the lectures is to cover topics in the following order. In the first lecture thedefining algebra of Supersymmetric Quantum Mechanics, the implications of this alge-bra for the spectra of the component parts of the SUSY Hamiltonian, the factorizationof the Schrödinger equation, the procedure for the elimination of the ground state of aHamiltonian, the procedure for the introduction of a new bound state below the groundstate of a given Hamiltonian and the procedure for generating a new Hamiltonian withunaltered spectrum will be discusssed using examples. Thus the first lecture will mainlybe concerned with showing that the existence of a SUSY partner to one-dimensionalHamiltonians implies a hierarchy of Hamiltonians with a special relationship betweenthe eigenvalues and eigenfunctions of the different members of the hierarchy.

In the second lecture the radial Schrödinger equation will be studied. The implica-tions of a SUSY partner to the radial Schrödinger equation will be used to differentiatebetween four types of SUSY transformations. In the first part of the third lecture the un-derlying supersymmetry of the Dirac equation for the Hydrogen atom will be discussed.In the second part of the third lecture the supersymmetry linking theN and theN + 1soliton solutions of the KdV equation will be discussed. The connection between the dif-ferent types of supersymmetric transformations of the radial Schrödinger equation andthe approach of the conventional inverse scattering theory will be more fully exploredin the fourth and fifth lectures. It will be shown how certain choices of pairs of SUSYtransformations lead to the results of the conventional inverse scattering theory basedon the Gelfand-Levitan and Marchenko equations. It will be shown that other choicesof pairs of SUSY transformations lead to new results not present in the standard inversescattering theory and produce singular potentials which have found a variety of applica-tions in Nuclear Physics. The fourth lecture will be concerned with the study of differentpairs of supersymmetric transformations and the different potentials that can be gener-ated by this procedure. In the fifth lecture the procedures discussed in the earlier lectureswill be generalized and some applications of the new aspects of inverse scattering the-ory arising from the singular potentials constructed using SUSY transformations of the

radial Schrödinger equation will be discussed.Throughout these lectures units in whichh = 1 and the massm= 1 will be used.

1.2. Supersymmetric quantum mechanics

SUSYQM is characterized by the existence of charge operatorsQi that obey thealgebra

{Qi ,Q j} = δi j H , i, j = 1,2, . . . ,N , (1)[Qi ,H] = 0 , (2)

whereH is the supersymmetric Hamiltonian,N is the number of generators and{,}denotes an anticommutator. Here we consider the simplest of such systems with twooperatorsQ1 andQ2. In terms ofQ = (Q1 + iQ2)/

√2 and its Hermitian adjointQ† =

(Q1− iQ2)/√

2 the algebra governing this supersymmetric system is characterized by

H = {Q,Q†} , Q2 = 0 , Q†2 = 0 . (3)

From these equations it is clear that

[Q,H] = 0 , [Q†,H] = 0 . (4)

i.e., the charge operatorQ is nilpotent and commutes with the HamiltonianH. A simplerealization of the algebra defined in Eq. (3) can be achieved by considering

Q =(

0 0A− 0

), Q† =

(0 A+

0 0

), (5)

whereA− is an operator andA+ is its adjoint. It is clear that with this constructionQ2 = 0automatically. Eqs. (3) and (5) lead to the supersymmetric Hamiltonian

H =(

A+A− 00 A−A+

). (6)

Since

Q

(α0

)=

(0

A−α

), Q†

(0β

)=

(A+β

0

), (7)

we can say that the operatorsQ andQ† induce transformations between the ‘bosonic’sector represented byα and the ‘fermionic’ sector represented byβ . We may alsointerpretH in the following way: the scalar HamiltonianH = A+A− has a partnerH =A−A+ such thatH andH are the diagonal elements of a supersymmetric HamiltonianH. Having demonstrated that aQ and anH can be constructed, we can switch to theoperator language of Quantum Mechanics to find out what the consequences of theexistence of a charge operator that commutes with the Hamiltonian are for the spectraof the two sectorsH andH. A+A− andA−A+ are both positive semi-definite operators

with eigenvalues greater than or equal to 0. Letψ be a normalized eigenstate ofH witheigenvalueE. Then

A+A−ψ = Eψ . (8)

Multiplication from the left byA− leads to

A−A+(A−ψ) = E(A−ψ) . (9)

If A−ψ 6= 0, we can infer thatE is also an eigenvalue ofA−A+. The correspondingnormalized eigenfunctionψ of H can be shown to be given by

ψ =1√E

(A−ψ) . (10)

The same reasoning may be applied starting from the eigenvalue equation forH insteadof H to investigate whether every eigenvalue ofH is also an eigenvalue ofH. If E is aneigenvalue ofH with eigenfunctionψ

A−A+ψ = Eψ , (11)

thenA+A−(A+ψ) = E(A+ψ) . (12)

Therefore, if A+ψ 6= 0 then E is also an eigenvalue ofH with the correspondingnormalized eigenfunction

ψ =1√E

(A+ψ) . (13)

In view of the above relationships, three possibilities may be distinguished from eachother.

(a) If there is a normalizable eigenstate ofH such thatA−ψ(0) = 0, thenA+A−ψ(0) = 0andψ(0) corresponds to the ground state with eigenvalueE(0) = 0. Conversely, for theeigenvalueE(0) = 0 the vanishing expectation value ofA+A− in the ground state impliesthat A−ψ(0) = 0. Under these circumstances,H has no normalizable eigenstate withE = 0, i.e. there can be no normalizable state withA+ψ = 0. The ground state eigenvalueof H is non-zero. All eigenvalues other than the ground state eigenvalue ofH are alsoeigenvalues ofH and all eigenvalues ofH are also eigenvalues ofH. The resultingspectral mapping is shown in Fig. 1(a).

(b) If there is a normalizable eigenstate ofH such thatA+ψ(0) = 0, thenA−A+ψ(0) =0 and ψ(0) corresponds to the ground state ofA−A+ with eigenvalueE(0) = 0. Therecannot be a normalizable state ofA+A− with eigenvalue zero that satisfiesA−ψ = 0.The ground state ofH has non-zero eigenvalue. All eigenvalues other than the groundstate ofH are also eigenvalues ofH and all eigenvalues ofH are eigenvalues ofH. Thisleads to the spectral mapping shown in Fig. 1(b).

(c) If there is no normalizable eigenstate ofH or H such that eitherA−ψ = 0 orA+ψ = 0, then the spectra of bothH andH begin at positive values. Every eigenvalueof H is also an eigenvalue ofH andvice versa. The resulting spectral mapping is shownin Fig. 1(c).

FIGURE 1. Schematic diagram of the possible allignment of eigenvalues of the operatorsH1 = H =A+A− andH2 = H = A−A+.

In each of the three cases the eigenfunctions ofH andH for a common eigenvalueEare linked in the manner indicated below:

ψ(E) = exp(iφ)(E)−12 A−ψ(E) ,

ψ(E) = exp(−iφ)(E)−12 A+ψ(E) , (14)

in whichφ is an arbitrary phase whose significance will become clear in later discussion.The ladder structure of the eigenvalue spectrum shown in Fig. 1 and the intertwiningrelationships between the eigenfunctions given above are characteristic hallmarks ofsupersymmetric systems in one dimension and serve as signatures by which the presenceof an underlying supersymmetry may be inferred. In the early works on SUSYQM([12, 15]) operators of the form

A± =(± d

dx+U(x)

), (15)

were considered in whichU(x) was considered to be a known function ofx. Thisassumption restricts the applicability of SUSYQM to a limited class of problems. Inthe next section it will be shown that it is not necessary to make any assumptions aboutU and thatU itself may be generated from the solutions of the Schrödinger equationin one dimension. Such a generalization extends the applicability of SUSYQM to allone-dimensional problems which have a ground state.

1.3. Factorization of the Schrödinger equation

The Schrödinger equation in one dimension is governed by the Hamiltonian

H =−12

d2

dx2 +V(x) , (16)

where V is the potential.H can be factorized in the form

H = A+A−+ ε , A± =1√2

(± d

dx+U

), (17)

whereε is an undetermined constant, provided that the unknown functionU satisfies(

dUdx

+U2)

= 2(V− ε) . (18)

This is a nonlinear equation with a family of solutions. One member of the family isgiven by

U =1

ψ(x,ε)ddx

ψ(x,ε) , (19)

whereψ(x,ε) is a solution of the Schrödinger equation at energyE = ε, i.e.

Hψ(x,ε) = εψ(x,ε) . (20)

SincedUdx

=1ψ

d2ψdx2 −

1ψ2

(dψdx

)2

, (21)

it is easy to verify that Eq. (19) satisfies Eq. (18). It is clear that this argument is validonly if ψ(x,ε) is non-vanishingi.e. ψ(x,ε) is nodeless. It can be shown that the generalsolution to Eq. (18) is given by the one-parameter family of solutions

U(x,ε,λ ) =ddx

lnψ(x,ε)+1/ψ2(x,ε)

λ +∫ xdz/ψ2(z,ε)

, (22)

whereλ is an arbitrary parameter. Every choice ofε and the correspondingψ(x,ε)leads to a possible factorization ofH in the formH = A+(ε)A−(ε)+ ε. The choice offactorization energyε and the selection of a value forλ must clearly be motivated bythe particular circumstances of a given problem and by physical considerations. If weconsider a Hamiltonian with a ground state energyE(0), then the requirement thatA+A−be a positive definite operator can be met only if the energyε is chosen to beε ≤ E(0).We consider the case when the factorization energyε = E(0) next.

1.4. Factorization energyε equals the ground state energyE(0)

With the choice ofE(0) as the factorization energy, the ground state eigenfunctionψ(x,E(0)) is nodeless and vanishes in the asymptotic region. The requirement thatU(x)in Eq. (22) should not be divergent leads to the choiceλ = ∞, giving

U(x) =ddx

lnψ(x,E(0)) ,

A±(E(0)) =1√2

[± d

dx+

ddx

lnψ(x,E(0))]

,

H = A+(E(0))A−(E(0))+E(0) . (23)

It is clear thatA+A− has a spectrum beginning at0, with a ground state which satisfiesA−ψ(0) = 0 with ψ(0) = ψ(x,E(0)). The analysis in §1.3 can now be used by consideringthe partner Hamiltonian

H = E(0) +A−(E(0))A+(E(0)) = H +[A−(E(0)),A+(E(0))] , (24)

corresponding to the potential

V(x) = V(x)− d2

dx2 lnψ(0) . (25)

H andH have their spectra aligned as in Fig. 1(a).H has no eigenstate corresponding tothe ground state ofH and all the excited states ofH are degenerate with the eigenstatesof H. The eigenfunctions of the two Hamiltonians are linked in the form

ψ(x,E) = (E−E(0))−12 A−(E(0))ψ(x,E) ,

ψ(x,E) = (E−E(0))−12 A+(E(0)) ψ(x,E) , (26)

by choosing the phaseφ in Eq. (14) to be zero. These equations are valid not only whenE is one of the discrete eigenvalues ofH , E = E( j)( j 6= 0), but also whenE lies in thecontinuous part of the spectrum. WhenE lies in the continuous part of the spectrumof H the above equations can be used to find a relation between the transmissioncoefficients in the potentialsV(x) andV(x) at energyE since the asymptotic form ofthe wavefunction for potentialV at energyE implies a definite asymptotic form for thewavefunction forV at the same energy. This procedure will be illustrated by consideringthe phase shift for the solutions of the radial Schrödinger equation in the second lecture.Since the above anlysis is valid for any one-dimensional HamiltonianH1 with the groundstate with energyE(0)

1 and wave functionψ(0)1 the process of finding a supersymmetric

partner can be iterated to generate the hierarchy of Hamiltonians given by

Hn(x) =−12

d2

dx2 +Vn(x)≡ A+n A−n +E(0)

n = A−n−1A+n−1 +E(0)

n−1 , n = 2,3, . . . , (27)

where

A±n (x) =1√2

[± d

dx+

ddx

lnψ(0)n (x)

], n = 1,2, . . . ,

Vn(x) = Vn−1(x)− d2

dx2 lnψ(0)n−1(x) , n = 2,3, . . . , (28)

in whichE( j)n andψ( j)

n are the eigenenergies and eigenfunctions ofHn with the propertythat

E(m)n = E(m+1)

n−1 = . . . = E(m+n−1) ,

ψ(m)n = [E(m)

n −E(0)n−1]

− 12 A−n−1ψ(m+1)

n−1 ,

ψ(m+1)n−1 = [E(m)

n −E(0)n−1]

− 12 A+

n−1ψ(m)n , n = 2,3, . . . , m= 0,1,2, . . . . (29)

FIGURE 2. Schematic diagram of the eigenvalue spectra of the Hamiltonians in the hierarchyHn. Thenumber of bound states ofH1 is arbitrarily chosen to be 5.

A pictorial representation of the eigenvalue correspondence of the Hamiltonian hierar-chy is given in Fig. 2.

The equations given above show that the excited states ofV1 can be obtained from theground states of the hierarchyVn. The simple harmonic oscillator, the particle in a box,the radial equation for a definite partial wave for the Coulomb potential and the Morsepotential are all examples where the potentials corresponding to the Hamiltonians in thehierarchy can be analytically worked out [17]. One nontrivial example of such an exactlysolvable hierarchy will be discussed next.

1.5. Attractive sech2 potential

LetV1 =−λ1sech2x , λ1 > 0 . (30)

Since this potential is attractive in all space−∞≤ x≤∞ it will support atleast one boundstate irrespective of the strength of the potential. In terms of the parameter

Q1 =(

2λ1 +14

) 12

≥ 12

, (31)

the spectrum of this potential is given by [20]

E(m)1 =−1

2

[Q1−

(m+

12

)]2

, m= 0,1,2, . . . ,N≤Q1− 12

. (32)

The potential in Eq. (30) supports a finite number(N +1) of bound states. The groundstate wavefunction

ψ(0)1 (x)∼ sech(Q1− 1

2)x , (33)

leads to

V2(x) =−(

λ1 +12−Q1

)sech2x . (34)

Inspection of this equation shows that(i) if λ1 > 1, thenV2(x) is an attractive sech2x potential;(ii) if λ1 = 1, thenV2(x) vanishes andH2 is a free particle Hamiltonian;(iii) if λ1 < 1, thenV2(x) is a repulsive potential and corresponds to a sech2x barrier.

It is easy to show that the parameter correspoding toQ1 for V2 is

Q2 =[

14

+2

(λ1 +

12−Q1

)] 12

= Q1−1 . (35)

The spectrum ofH2 is then given by

E(m)2 =−1

2

[Q2−

(m+

12

)]2

, (36)

which satisfies the conditionE(m)2 = E(m+1)

1 . Iteration of this argument shows that theHamiltonian hierarchy corresponds to a sequence of shape-invariant potentials withsuccessively decreasing strengths. It is easy to show that

Vn(x) =−λnsech2x , Qn =(

2λn +14

) 12

= Qn−1−1 . (37)

If N = Q1− 12 thenVN+1(x) vanishes. IfN < Q1− 1

2 < N+1 thenVN+2 corresponds toa sech2 barrier given by

VN+2(x) =12

(Q1−N− 1

2

)(N+

32−Q1

)sech2x . (38)

We have shown that by choosing the factorization energyε to be the ground stateenergy it is possible to generate a new HamiltonianH2 without an eigenstate corre-sponding to the ground state ofH1 but retaining the rest of the spectrum ofH1. It hasbeen demonstrated that this procedure may be iterated to generate a Hamiltonian hierar-chy with spectra aligned as in Fig. 2. In the next subsection we examine other possiblefactorizations.

1.6. Factorization energyε less than the ground state energy

When the factorization energyε in Eq. (17) is less than the ground state energy ofH the solutionψ(x,ε) of Hψ = εψ is not a normalizable solution eventhoughψ(ε)is still a solution ofA−ψ(ε) = 0. The lack of normalizability ofψ(ε) means thatA+(ε)A−(ε) cannot have zero as an eigenvalue and the spectrum ofA+A− beginsat positive values. The analysis in §1.3 shows that whenA+A− has no normalizableeigenstate with eigenvalue zero, it is possible forA−A+ to have a spectrum beginningat eigenvalue zero. ForA−A+ to have a normalizable state with eigenvalue zero, thesolutionψ of A+(ε)ψ = 0 must be normalizable. The solution of

[ddx

+ddx

lnψ(x,ε)]

ψ(x,ε) = 0 , (39)

i.e.

ψ(x,ε) =1

ψ(x,ε), (40)

shows that if the unnormalizable solutionψ(x,ε) of the HamiltonianH is chosen insuch a way that(ψ)−1 is normalizable, thenψ(x,ε) is normalizable andA−A+ has aspectrum beginning at eigenvalue zero. Therefore

H = A−(ε)A+(ε)+ ε , ε < E(0) , (41)

has a ground state at energyE(0) with a ground state eigenfunctionψ(0)(x,ε) = ψ(x,ε).Therefore,H has a ground state eigenvalue below the ground state ofH while all theother eigenvalues ofH are degenerate with the eigenvalues ofH. This corresponds tothe level scheme shown in Fig. 1(b). Hence when(ψ)−1 is normalizable

H = H− d2

dx2 [lnψ(x,ε)] , (42)

has ground state

E(0) = ε < E(0) , ψ(0)(x,ε) =1

ψ(x,ε), (43)

and excited states with

E(m+1) = E(m) , m= 0,1,2, . . . ,

ψ(m+1) = −(

E(m)− ε)− 1

2A−ψ(m) ,

ψ(m) = −(

E(m)− ε)− 1

2A+ψ(m+1) ,

A±(ε) =1√2

[± d

dx+

ddx

lnψ(x,ε)]

. (44)

The phase factorφ in Eq. (14) has been chosen to beπ. Having chosenφ to be zerofor the case of elimination of a state in §1.5, the requirement that adding a state by atransformation and subsequently eliminating the same state by another transformationshould give back the original transformation, fixes the phase factor for the case ofthe addition of a state to beπ. If ε < E(0), but the unnormalizable solutionψ(x,ε)does not lead to a normalizable(ψ)−1 and the second derivative oflnψ(x,ε) is wellbehaved, in a sense to be defined shortly, then neitherA+(ε)A−(ε) nor A−(ε)A+(ε)has a normalizable eigenstate with eigenvalue zero. We denote such a solutionψ by ξ .ThereforeA+A− andA−A+ have identical spectra as depicted in Fig. 1(c). Then

H = A−(ε)A+(ε)+ ε = H− d2

dx2 lnξ (x,ε) , (45)

has a spectrum identical to that ofH. The relations between the eigenvalues and theeigenstates are given by

E(m) = E(m) , m= 0,1,2, . . . ,

ψ(m) = exp(iφ)(

E(m)− ε)− 1

2A−ψ(m) ,

ψ(m) = exp(−iφ)(

E(m)− ε)− 1

2A+ψ(m) ,

A±(ε) =1√2

[± d

dx+

ddx

lnξ (x,ε)]

. (46)

The phase factorφ has been left undetermined. Furthermore, the non-normalizablesolutionsψ andψ for energyε are connected by

ψ(x,ε) =1

ψ(x,ε). (47)

In this section we assume that−∞≤ x≤ ∞ and postpone the discusion of0≤ r ≤ ∞ toa later section. It is necessary to make this distinction because the type of singularitiesof the potentialV that are physically admissible depends on the range of values of thevariablex. Potentials with singularities of the formr−2 are admissible for the radialproblem, but singularities of the formx−2 are inadmissible when−∞ ≤ x ≤ ∞. Thediscussion of the construction of a normalizableψ−1 depends on the spatial domain inwhichψ andV are defined. We now examine the question of the normalizability ofψ−1

when−∞ ≤ x≤ ∞. Let φ1(x,ε) be a nodeless solution ofHψ = εψ for ε < E(0). Theexistence of such a solution can be rigorously proved [21]. Another linearly independentsolution at the same energy is given by

φ2(x,ε) = φ1(x,ε)∫ x dz

φ21(z)

. (48)

The nodelessness ofφ1 guarantees that this integral is well defined. The general solutionat the energyε is given by

ψ(x,ε,α) = φ1 +αφ2 = φ1

(1+α

∫ x

−∞

dz

φ21

), (49)

in which the lower limit of the integral has been chosen to be−∞ andα is an arbitraryconstant. Let

β (ε) =(∫ ∞

−∞

dz

φ21(z,ε)

)−1

. (50)

It can be shown that for values ofα in the range−β < α < ∞, ψ will remain nodelessandψ is unnormalizable. It can also be shown that for−β < α < ∞, ψ−1 is singularityfree and normalizable. This range of values ofα then leads to a normalizableψ−1 andψ = ψ−1 corresponds to an eigenstate ofH as defined in Eq. (42) with ground state

eigenvalueE(0) = ε. It can be shown that for the limiting valuesα = −β andα = ∞even thoughψ−1 is unnormalizable the second derivative oflnψ is divergence free andfinite in the asymptotic region. These values ofα then lead toH defined in Eq. (45)with the same spectrum asH. But if α <−β thenψ will vanish for some finite value ofx as can be seen from Eq. (49) and the second derivative oflnψ(x,ε,α) then divergeswhenψ vanishes. Hence forα <−β , ψ(x,ε,α) does not lead to a physically acceptablepotentialV. The above analysis will be illustrated with examples in the next section.

1.7. Free particle, addition of bound state

Let V(x) = 0. H has only a positive energy spectrum. For negative energiesε =−γ2/2, the general solution ofHψ = εψ is given by

ψ(x,ε) = coshγx+α sinhγx . (51)

Thoughψ is unnormalizable, for values of the parameterα in the range|α | < 1 ψ isnodeless andψ−1 is normalizable. The family of potentials

V = V− d2

dx2 lnψ(x,ε) =− γ2(1−α2)(coshγx+α sinhγx)2 , |α|< 1 , (52)

therefore have a single bound state at energy

E(0) =−12

γ2 , (53)

with ground state eigenfunction

ψ(0) ∼ 1ψ

=1

coshγx+α sinhγx, |α |< 1 . (54)

For positive energies, Eq. (46) then gives

ψ(x,ε) =−[2(E− E(0))

]− 12(− d

dx+ γ

sinhγx+α coshγxcoshγx+α sinhγx

)ψ(x,E) . (55)

In the asymptotic region|x| → ∞, this equation becomes

lim|x|→∞

ψ(x,E) =−[2(E− E(0))

]− 12(− d

dx+ γ

)lim|x|→∞

ψ(x,E) . (56)

The α independence of this equation means that the transmission coefficient of thisfamily of potentialsV(x,E,α) are identical. This family of potentials is an example ofthe phase-equivalent family of Bargmann [22].

1.8. Simple harmonic oscillator, addition of bound state

The oscillator potential does not belong to the category of potentials that remain finitein the asymptotic region. Nevertheless, the oscillator example serves to clarify some ofthe discussion in the text. The harmonic oscillator Hamiltonian

H =−12

d2

dx2 +12

x2 , (57)

has the eigenvalue spectrum

E = (n+12) , n = 0,1,2, . . . . (58)

The even solution ofHψ = εψ for all energies can be written in series form [23] and isgiven by

φ1(x,ε) =(

1+δx2

2!+δ (4+δ )

x4

4!+δ (4+δ )(8+δ )

x6

6!+ . . .

)e−x2/2 , (59)

whereδ = 1−2ε . (60)

For energies below the ground state of the oscillatorε < E(0) = 12, δ > 0, which

guarantees thatφ1 is positive definite. Thusφ1 is a nodeless unnormalizable solutionfor ε < 1

2. The linearly independent solution

φ2(x,ε) = φ1

∫ x

0

dz

φ21(z)

, (61)

can also be written in series form as

φ2(x,ε) =(

1+(δ +2)x2

3!+(δ +2)(δ +6)

x4

5!+ . . .

)xe−x2/2 . (62)

φ2 vanishes atx= 0 but the series within the parentheses is positive definite whenε < 12.

Both φ1 andφ2 may be expressed in terms of standard parabolic cylinder functions. Thegeneral solution at energyε is then given by

ψ(x,ε) = φ1(x,ε)+αφ2(x,ε) . (63)

In terms of the parameter

limx→∞

φ1

φ2= β (ε) =

(∫ ∞

0

dz

φ21(z,ε)

)−1

= 2Γ(3

4− 12ε)

Γ(14− 1

2ε), (64)

which arises from asymptotic formulae for the parabolic cylinder functions [23], for|α|< β

ψ(x,ε,α) = φ1(x,ε)(

1+α∫ x

0

dz

φ21(z,ε)

), (65)

FIGURE 3. The phase-equivalent potentialsV(x,ε,α) are shown forε = −1/2. The value ofα isindicated below each curve. The harmonic oscillator potentialV(x) = x2/2 is also shown as a brokencurve. The potentials shown in figure by full curves have identical spectra with ground state atE(0) =−1/2as indicated by the horizontal broken line. The rest of the spectrum is identical to that of the harmonicoscillator.

is nodeless andψ−1 is normalizable. The family of Hamiltonians

H = H− d2

dx2 lnψ(x,ε,α) , |α |< β , (66)

therefore have identical spectra given by

E(0) = ε <12

, E(m) = m− 12

, m= 1,2, . . . . (67)

This family of potentials is another example of the phase-equivalent family of Bargmann[22]. Since the energyE(0) is arbitrary as long asE(0) < 1

2, the above equations give arecipe for constructing anharmonic potentials with spectra defined by Eq. (67).

Using the series expansion forφ1 andφ2 the potentialsV(x,ε,α) have been calculatedfor a range of values ofε andα < β (ε). Fig. 3 showsV for ε =−1/2 andα in the rangeof values0 < α < 2/

√π. The results for positive values ofα are shown. The potential

for the corresponding negative value ofα may be obtained by mirror reflection aboutthex-axis. Forα = 0, V(x) = x2/2−1 is a shifted oscillator. This is the only value ofα for whichV is invariant under the parity transformation. Thus by imposing a specific

(a) (b)

FIGURE 4. The potentialV(x,ε,α) for α = 0 and (a)ε = 0.45, (b) ε = 0.0, The harmonic oscillatorpotential is indicated as a broken curve. The ground state ofV at E = 0.5 is indicated by a full horizontalline and the ground state ofV at E = ε is indicated by a broken horizontal line. The rest of the spectrumof V is identical to that of the harmonic oscillator.

condition onV(x) a unique member of the family is obtained. Fig. 4 showsV(x,ε,0) fora range of values ofε for a fixed value ofα = 0. These figures show that for0 < ε < 1

2the ground state of the new potentialV lies inside the double well. This is an exampleof the general result that whenε lies below the ground state of a given potentialV, butε > Vmin(x) whereVmin is the minimum of the potential, the resulting partner potentialV(x) is necessarily a double well. It can be shown on general grounds that a double wellis necessary to accomodate the new level atε close to the first excited state ofV(x) atenergyE(0). Fig. 4 shows double well potentials whose spectrum is fixed by constructionto be of the form given in Eq. (67).

We next consider the limiting valuesα = β (ε) for whichψ(x,ε,β )−1 is unnormaliz-able. The value ofβ for a given value ofε may be found from Eq. (64). It can be shownthat the second derivative oflnψ is divergence free. Hence

V(ε) =x2

2− d2

dx2 lnψ(x,ε,±β (ε)) , (68)

has the spectrum

E(m) = m+12

, m= 0,1,2, . . . , (69)

which is identical to the spectrum of the harmonic oscillator. The eigenfunctions ofH may be given in terms of the oscillator eigenfunctions using the intertwining rela-tions involving theA± operators using the logarithmic derivative ofψ(x,ε,β (ε)). ThusHamiltoniansH(ε) for various values ofε < 1

2 which have spectra identical to the har-monic oscillator may be constructed. The potentialsV(x,ε,β (ε)) have been calculated

FIGURE 5. The potentialV(x,ε,β (ε)) for a range of values ofε. The ε value is indicated on eachcurve. The harmonic oscillator potential is shown as a broken curve. All the potentials shown in thisdiagram have spectra identical to that of the harmonic oscillator. The asymptotic values of the full curvesare given bylimx→∞ ∆V =−1, limx→−∞ ∆V = +1 where∆V = V(x)−V(x).

numerically and are shown in Fig. 5 for a range of values ofε and positive values ofβ (ε). The potentials for negativeβ may be obtained by mirror reflection.

1.9. Summary

It has been demonstrated that the algebra of supersymmetry can be used to find partnerHamiltonians to one-dimensional Hamiltonians. The flexibility in the choice of the part-ner Hamiltonian enables the identification of different types of supersymmetric pairings.A procedure for constructing Hamiltonians with either identical spectra or with identicalspectra apart from a missing ground state has been given. This procedure has been illus-trated with several examples. This recipe can be used to either add a new ground stateeigenvalue to, or eliminate the ground state of or maintain the same spectrum of a givenHamiltonian corresponding to a Schrödinger equation. This procedure may be repeatedagain and again in a suitable combination to generate hierarchies of Hamiltonians whosespectra are related to each other. By applying this procedure to the harmonic oscillator

anharmonic potentials whose spectra are identical to that of the harmonic oscillator orcontain a ground state lower in eigenvalue than the ground state of the harmonic oscil-lator have been constructed.

2. SUSYQM AND INVERSE SCATTERING THEORY

The radial Schrödinger equation corresponding to a definite partial wave is studied.The procedures for finding a new potential by eliminating the ground state of a givenpotential by adding a bound state below the ground state of a given potential and bygenerating the phase-equivalent family of a given potential using the supersymmetricpairing of the spectra of the operatorsA+A− andA−A+ are examined. Four differenttypes of transformations generated by the concept of a supersymmetric partner to a givenradial Schrödinger equation are identified and the modifications of the Jost functions forthe four transformations are classified. It is argued that the Bargmann class of potentialsmay be generated using suitable combinations of the four types of transformations.

2.1. Introduction

In the first lecture (§1) it was shown that by using the idea of a supersymmetric partnerto a Hamiltonian functionH of a single variable it is possible to find another HamiltonianH which had one of the following features: either (i) the complete spectrum ofH is madeup of all the eigenvalues ofH except the ground state ofH , or (ii) the complete spectrumof H is made up of all the eigenvalues ofH and in addition one further eigenvalue whichlies below the ground state ofH, or (iii) the spectrum ofH is identical to that ofH.It was shown that in all three cases the eigenfunctions ofH and H for the commoneigenvalues are connected by a linear differential operator. By repeated application ofthis procedure of either deleting an eigenvalue or adding an eigenvalue or maintainingthe same eigenvalues it is possible to generate Hamiltonians whose spectra bear adefinite relationship to each other. The inverse scattering theory can also accomplishthe same tasks through solving either the Gelfand-Levitan [24] or the Marchenko [25]equations [13, 26, 27]. The aim of this lecture is to elucidate the relationship betweenthe two approaches [28]. It will also be shown that the Bargmann class of potentials [22]may be genearted by the application of the concept of supersymmetric pairing.

The radial Schrödinger equation differs from the Schrödinger equation in the space[−∞,∞] in essential respects. The boundary conditions on the eigenfunctions and theallowed singularities of the potential are different in the two spaces[−∞,∞] and[0,∞].In this lecture the modifications introduced by switching fromx to r will be consideredfirst and then the modifications of the Jost function corresponding to four different typesof transformations will be studied.

2.2. The radial Schrödinger equation

We now consider the radial Schrödinger equation for a definite partial wave with theHamiltonian

H = −12

d2

dr2 +V(r) ,

V(r) =l(l +1)

2r2 +v(r) . (70)

The potentialV(r) is assumed to be regular, not singular. Specifically, the potentialsdiscussed here are restricted to be no more singular thanr−2 at the origin and decreasingat least as fast asr−2 asr → ∞.

In this lecture the term ‘normalization constant of the eigenfunction’ will be usedoften. This term has a specific meaning in the terminology of the inverse scatteringtheory. All bound state eigenfunctions are understood to be normalized to unity in theusual way to reflect the condition that the total probability of finding the bound particlesomewhere in space should be unity. However, in the inverse scattering method the term‘normalization constant’ is used in a specific sense. The regular solutionφ of the radialSchrödinger equation is defined to be a solution that satisfies the boundary condition

limr→0

φ(r,E, l) =r l+1

(2l +1)!!. (71)

The regular solution will grow exponentially asr →∞ whenE is not one of the eigenen-ergies. However, whenE is one of the eigenenergiesE(i) the bound state eigenfunction,which decreases exponentially asr → ∞, is proportional to the regular solution

ψ(r,E(i), l) = αφ(r,E(i), l) ,

∫ ∞

0ψ2dr = 1 . (72)

It is this proportionality constantα that corresponds to the ‘normalization constant’referred to in the inverse scattering method. Throughout this lecture the term ‘normal-ization constant’ will be used in the sense in which it is used in the inverse scatteringtheory. The term ‘normalizable’ will, however, be used in the usual sense of the word,i.e.,

∫ ∞0 ψ2dr is finite. In view of the different types of transformations of the radial equa-

tion that will be discussed the following notations will be adopted. The eigenfunctionsof H defined in Eq. (70) are denoted byψ(i) for the discrete states at energiesE(i), thephase shifts for the continuum statesψ(r,E) for positive energiesE = 1

2k2 are denotedby δ (l ,k) and the Jost function byF(l ,k). The potentials, eigenstates, phase shifts andJost function after the supersymmetric transformation are denoted by adding a tilde,ψ(r,E), for example. The different types of transformations are distinguished by addinga suffix,ψ1(r,E), for example. We adopt the notation thatψ(m)(r) is an abbreviation forψ(r,E(m)).

2.3. Jost function

In scattering theory the S-matrix may be constructed from the Jost function [29]. Theintegral representation of the Jost function for a potentialv(r) with N bound states atenergiesE = E(i) and phase shiftsδ (l ,k) at energiesE = 1

2k2 for angular momentumlis given by (see Chadan and Sabatier [30], for example)

F(l ,k) =N

∏i=1

(1− E(i)

E

)exp

(−2

π

∫ ∞

0

δ (l , p)pdpp2−k2

). (73)

The phase of the Jost function is−δ (l ,k) while the modulus is given by

|F(l ,k)|=N

∏i=1

(1− E(i)

E

)exp

(−2

πP

∫ ∞

0

δ (l , p)pdpp2−k2

), (74)

where the symbolP stands for principal value. The spectral density for positive energiesis given by

dP(E)dE

=E(l+ 1

2)

π|F(l ,k)|−2 . (75)

Knowledge of the phase shifts for all positive energies, the bound state energiesE(i) andthe normalization constantsC(i) associated with each of the bound states enables thecomplete determination of the potentialV(r).

2.4. Elimination of the ground state

By the methods discussed in §1 it can be shown thatH defined by Eq. (70) has asupersymmetric partner

H1 = H− d2

dr2 lnψ(0)(r) . (76)

Sincelimr→0

ψ(0(r)∼ r l+1 , (77)

H1 corresponds to the potential

V1(r) =(l +1)(l +2)

2r2 +v(r)− d2

dr2 ln

(ψ(0)(r)

r l+1

), (78)

where the singularity at the origin has been separated to show the behaviour near theorigin. It can be established that

limr→0

V1(r) =(l +1)(l +2)

2r2 ,

limr→∞

V1(r) =l(l +1)

2r2 . (79)

V1 has no normalizable state with eigenvalueE(0) and therefore the ground state of V ismissing from the spectrum of eigen values forV1. All the other eigenvalues of the twopotentials are the same. The eigenfunctions are related by

ψ(m)1 = (E(m)−E(0))−

12 A−1 ψ(m) , m= 1,2, . . . ,

A−1 =1√2

[− d

dr+

ddr

lnψ(0)(r)]

. (80)

Extension of the above eigenfunction relation to positive energy states and use of theasymptotic forms

limr→∞

ψ(r,E) ∼ sin

(kr− 1

2lπ +δ (l ,k)

),

limr→∞

ψ(0)(r) ∼ exp(−γ(0)r) , (81)

then gives

limr→∞

ψ1(r,E)∼ sin

(kr− 1

2lπ + δ1(l ,k)

), (82)

where

δ1(l ,k) = δ (l ,k)− π2− tan−1

(γ(0)

k

), E =

12

k2 , E(0) =12

(γ(0)

)2. (83)

The phase shift relation is consistent with the observation that whenr → 0 the singularityof the potentialV1 corresponds tol → l + 1 which implies an increased repulsion andtherefore the phase shift should decrease. In the limitk→ 0 the phase shift decreases byπ which is the correct limit when a bound state has been eliminated. In the limitk→ ∞the phase shift decreases byπ/2. Eqs. (73), (74), (80) and (83) enable the establishmentof a relationship between the Jost functions for the potentialsV1 andV. The phaseshift relation given in Eq.(83) may be used to compare the Jost functions for the twopotentials. Using the integral relation [31]

4π

P∫ ∞

0

cot−1(p/γ)p2−k2 pdp= ln

(1+

γ2

k2

), (84)

we can establish that the Jost functions for the two potentials are related by

F1(l ,k)F(l ,k)

=k

k− iγ(0) . (85)

2.5. Addition of a bound state

The potentialV2 with a ground state atE =−12γ2 < E(0), i.e., below the ground state

of V in addition to sharing all the eigenvalues ofV can be constructed by the methods

of §1. Since the potential in the radial equation can have singularities of the formr−2

the equations of §1 must be recast in an appropriate form. The regular solution in thepotentialV at energyE denoted byφ satisfies

limr→0

φ =r l+1

(2l +1)!!, lim

r→∞φ ∼ exp(γr) . (86)

Since the energyE is below the ground state ofV , φ is nodeless forr > 0 and may bechosen to be positive definite forr > 0. The linearly independent solution can be takento be

η(r) = φ(r)∫ ∞

r

dzφ2(z)

. (87)

It is easy to show that

limr→0

η(r)∼ r−l , limr→∞

η(r)∼ exp(−γr) . (88)

The functionη is one of the Jost solutions (see [29], for example) defined by a boundarycondition in the asymptotic region. For energiesE < E(0), η is also a nodeless functionand is positive definite. WhenE is not only less thanE(0) but also less than the absolutemimimium of the potentialV then the positivity of(V − E) guarantees thatφ andηare monotonically growing functions ofr in the directionsr = 0→ ∞ andr = ∞ → 0,respectively. WhenVmin < E < E(0), φ and η are no longer monotonically growingfunctions but nevertheless remain nodeless. These assertions on the behaviour ofφ andη may be rigorously proved. The function

ψ = φ cosα +η sinα , (89)

is also a nodeless function when0< α < 12π andψ−1 is a normalizable function for this

range of values ofα since

limr→0

1ψ

= limr→0

1η sinα

∼ r l , (90)

and

limr→∞

1ψ

= limr→∞

1φ cosα

∼ exp(−γr) . (91)

By the methods of §1 it is easy to infer that when0 < α < 12π

V2(r) = V(r)− d2

dr2 lnψ(r, E,α) , (92)

with

ψ(r, E,α) = φ(r, E)(

cosα +sinα∫ ∞

r

dz

φ2(z, E)

), (93)

has a ground state atE(0)

2 = E , (94)

with eigenfunction

ψ(0)2 (r, E,α) =

1

ψ(r, E,α). (95)

All the other eigenvalues ofV2 are identical to the eigenvalues ofV. The other eigen-functions ofV2 are given by

ψ(m)2 = −(E(m)− E)−

12 A−2 ψ(m) , m= 1,2, . . . ,

A−2 (E,α) =1√2

[− d

dr+

ddr

lnψ(r, E,α)]

. (96)

The potentialV2 may be written in the form

V2(r, E,α) =l(l −1)

2r2 +v(r)− d2

dr2 ln(r l ψ(r, E,α)) , (97)

where the singularity at the origin has been separated to show the behaviour near theorigin. It can be established that

limr→0

V2 =l(l −1)

2r2 , limr→∞

V2 =l(l +1)

2r2 . (98)

Extension of Eq. (96) to positive energy states and use of the asymptotic forms

limr→∞

ψ(r,E) ∼ sin

(kr− 1

2lπ +δ (l ,k)

),

limr→∞

ψ(r, E,α) ∼ exp(γr) , (99)

then gives

limr→∞

ψ(r, E,α)∼ sin

(kr− 1

2lπ + δ (l ,k)

), (100)

where

δ (l ,k) = δ (l ,k)+π2

+ tan−1(

γk

). (101)

The phase shift relation is consistent with the observation that whenr → 0 the singularityof the potentialV2 corresponds tol → l −1, which implies a decrease in the repulsionand therefore the phase shift should increase. In the limitk→ 0 the phase shift increasesby π which is the correct limit when the number of bound states increases by one. In thelimit k→ ∞ the phase shift increases byπ/2. The equations given above show that allmembers of the familyV2(r, E,α) lead to identical phase shifts for the same energyEwhen0 < α < 1

2π. Furthermore, since

limr→0

ddr

lnψ(r, E,α) =− lr

, (102)

Eq.(96) shows that for a fixed principal quantum numberm, limr→0 ψ(r, E,α) is inde-pendent ofα and therefore the excited states ofV2(r, E,α) for variousα have identicalnormalizations. However, the normalized ground state eigenfunction

ψ(0)2 (r, E,α) =

(sinα cosα)12

φ (cosα +sinα∫ ∞

r dz/φ2(z)),

∫ ∞

0

[ψ(0)

2

]2dr = 1 , (103)

shows that

limr→0

ψ(0)2 (r, E,α)∼

(sinαcosα

) 12

r l . (104)

Hence the ground state eigenfunction of the family of potentialsV2(r, E,α) have differ-ent normalizations,i.e., different proportionalities to the regular solution, although theybelong to the same eigenvalue. It has been shown that the phase shifts, the eigenvaluesand the normalization constants of the excited states are identical for all members of thefamily of potentialsV2(r, E,α),0 < α < 1

2π, while the ground state eigenfunctions be-

longing to the eigenvalueE(0)2 = E have different normalization constants for different

values ofα. Clearly the familyV2(r, E,α) is an example of the phase equivalent familywhich was discussed by Bargmann [22].

The phase shifts and the bound state energies ofV andV2 enable the comparison ofthe Jost functions. From Eqs. (73) and (102) it is easy to show that

F2(l ,k)F(l ,k)

=k− iγ

k. (105)

2.6. Boundary values ofα and equivalent potentials

When the parameterα lies outside the range0 < α < 12π, the eigenfunctionψ in

Eq. (89) does not lead to a normalizableψ−1. When−π < α < 0 or π > α > 12π, ψ

vanishes at some finite value ofr because eithersinα or cosα assumes negative valuesand

∫ ∞r dz/φ2 can take all values from0 to ∞. If ψ vanishes at a finite value ofr then

the second derivative oflnψ diverges at this point. This would then lead to a singularV.However, the critical valuesα = 0 andα = 1

2π must be studied separately.(a) Whenα = 0

ψ(r, E,0) = φ , limr→0

ψ ∼ r l+1 , limr→∞

ψ ∼ exp(γr) . (106)

The vanishing value ofψ at r = 0 shows thatψ−1 is not normalizable, but

limr→0

d2

dr2 lnφ ∼−(l +1)r2 , lim

r→∞

d2

dr2 lnφ ∼ 0 . (107)

The positivity of φ guarantees that there are no singularities in the second derivativeof lnφ for r > 0. These conditions ensure that it is possible to find a non-singular

supersymmetric partner toV. It can be shown that

V3(r, E) =(l +1)(l +2)

2r2 +v(r)− d2

dr2 ln

(φ(r, E)

r l+1

), (108)

and that

limr→0

V3(r, E) =(l +1)(l +2)

2r2 , limr→∞

V3(r, E) =l(l +1)

2r2 . (109)

The eigenvalue spectrum ofV3 is identical to that ofV and the new eigenfunctions are

ψ(m)3 = (E(m)− E)−

12 A−3 ψ(m) ,

A−3 (E) =1√2

[− d

dr+

ddr

lnφ(r, E)]

. (110)

The phase shifts in the potentials are related by

δ3(l ,k) = δ (l ,k)− π2

+ tan−1(

γk

). (111)

The phase shift relation is consistent with the observation that whenr → 0 the singularityof the potentialV3 corresponds tol → l + 1, which implies an increased repulsion andtherefore the phase shift should decrease. In the limitk→∞ the phase shift decreases byπ/2. In the limit k→ 0 the phase shift is unchanged which is the correct limit since thetwo potentials have the same number of bound states. The family of potentialsV3(r, E)for different values ofE < E(0) have identical spectra but different phase shifts for thesame energy. The Jost functions for the potentialsV andV3 can be shown to be relatedin the manner

F3(l ,k)F(l ,k)

=k

k+ iγ. (112)

(b) Whenα = 12π

ψ(r, E,12

π) = η , limr→0

∼ r−l , limr→∞

ψ ∼ exp(−γr) . (113)

Henceα = 12π does not lead to a normalizableψ−1, becauseψ−1 diverges asr → ∞.

However,

limr→0

d2

dr2 lnη =lr2 , lim

r→∞

d2

dr2 lnη ∼ 0 . (114)

These conditions together with the absence of any singularities in the second derivativeof lnη ensure that a singularity-free supersymmetric partner toV may be constructed. Itcan be shown that

V4(r, E) =l(l −1)

2r2 +v(r)− d2

dr2 ln(

r l η(r, E))

, (115)

and that

limr→0

V4(r, E) =l(l −1)

2r2 , limr→∞

V4(r, E) =l(l +1)

2r2 . (116)

The eigenvalue spectrum ofV4 is identical to that ofV and the eigenfunctions are relatedby

ψ(m)4 = −(E(m)− E)−

12 A−4 ψ(m) ,

A−4 (E) =1√2

[− d

dr+

ddr

lnη(r, E)]

. (117)

The phase shifts in the potentials are related by

δ4(l ,k) = δ (l ,k)+π2− tan−1

(γk

). (118)

The phase shift relation is consistent with the observation that whenr → 0 the singularityof the potentialV4 corresponds tol → l − 1, which implies a decreased repulsion andtherefore the phase shift should increase. In the limitk→ ∞ the phase shift increases byπ/2. In the limit k→ 0 the phase shift is unchanged which is the correct limit since thetwo potentials have the same number of bound states. The family of potentialsV4(r, E)for different values ofE < E(0) have identical spectra but different phase shifts for thesame energy. The Jost functions for the potentialsV andV4 can be shown to be relatedin the manner

F4(l ,k)F(l ,k)

=k+ iγ

k. (119)

2.7. Summary

It has been shown that by a suitable factorization of the radial Schrödinger equation, itis possible to discover an underlying supersymmetric algebra. This algebra may be usedto generate four different types of transformations of the radial Schrödinger equation.The four transformations may be classified as follows.

(1) T1 is a transformation that eliminates the ground state[E(0),ψ(0)] of the potentialV(r), leaves the rest of the spectrum of eigenvalues unaltered, leaves the angular mo-mentum component of the potential in the regionr →∞ unaltered and alters the singularbehaviour nearr → 0 to a centrifugal potential corresponding to angular momentum(l + 1). T1 also changes the Jost function corresponding toV by a multiplicative factork/(k− iγ(0)) whereγ(0) = [−2E(0)]

12 . The new eigenfunctions in the potential (78) are

given by Eq. (80).(2) T2 is a transformation that adds a bound stateE(0)

2 , ψ(0)2 below the ground state

of the potentialV(r), leaves the rest of the spectrum of eigenvalues unaltered, leavesthe angular momentum component of the potential in the regionr → ∞ unaltered andalters the singular behaviour nearr → 0 to a centrifugal potential corresponding toangular momentum(l − 1). T2 also changes the Jost function corresponding toV by

a multiplicative factor(k− iγ(0)2 )/k whereγ(0)

2 = [−2E(0)2 ]

12 . The new eigenfunctions in

the potential in Eqs. (92) and (93) are given by Eqs. (95) and (96).(3) T3 is a transformation that maintains the eigenvalue spectrum ofV unaltered,

leaves the angular momentum component of the potential in the regionr → ∞ unalteredand alters the singular behaviour nearr → 0 to a centrifugal potential corresponding toangular momentum(l +1). T3 also alters the Jost function forV by a multiplicative factork/(k+ iγ) where γ = [−2E]

12 and E < E(0). The new eigenfunctions in the potential

(108) are given by Eq. (110).(4) T4 is a transformation that maintains the eigenvalue spectrum ofV unaltered,

leaves the angular momentum component of the potential in the regionr → ∞ unalteredand alters the singular behaviour nearr → 0 to a centrifugal potential corresponding toangular momentum(l−1). T4 also alters the Jost function forV by a multiplicative factor(k+ iγ)/k where γ = [−2E]

12 and E < E(0). The new eigenfunctions in the potential

(115) are given by Eq. (117).In the one-dimensional case−∞ < x< ∞, the singularities in the potential atx= 0 are

not permitted and the physical eigenfunctions are defined by boundary conditions at±∞.In the case of the radial equation the boundary conditions on the eigenfunctions atr = 0andr = ∞ are different. All four types of transformations listed above have analogues inthe space[−∞,∞] but with the difference that no singularities of the typex−2 should beintroduced by the transformations because of the boundary conditions usually imposedon ψ(x). In thex spaceV3 andV4 arise from considering the limiting valuesα = ∞ andα =−β discussed in §1.6.

2.8. Relationship to Bargmann potentials

It has been shown above that each of the transformationsT1− T4 corresponds toa multiplication of the Jost function by a specific rational function ofk. By repeatedapplication of a combination of the four types of transformations in an appropriate orderthe Jost function ofV can be modified by any rational function ofk. The generation of theBargmann class of potentials [7] corresponds to such a modification of the Jost function.Therefore it is clear that the Bargmann class of potentials may be generated by a suitablecombination ofT1, T2, T3, andT4. For example, the multiplication of the Jost function bya factor(k+ ib)/(k+ ia) can be broken down into two steps, multiplication by(k+ ib)/kfollowed by a further multiplication byk/(k+ ia), corresponding to application ofT4followed byT3. The physically acceptable Jost functions must satisfy the condition

limk→∞

F(l ,k) = 1 , (120)

and the symmetry relationF(l ,−k?) = [F(l ,k)]? . (121)

The modifications ofF introduced byT1-T4 clearly satisfies these conditions.

3. SUPERSYMMETRY AND THE DIRAC EQUATION FOR ACENTRAL COULOMB FIELD

It is shown that the methods of Supersymmetric Quantum Mechanics can be used toobtain the complete energy spectrum and eigenfunctions of the Dirac equation for anattractive Coulomb potential.

3.1. The Dirac equation for the Coulomb potential

The Dirac equation for the electron in an attractive central Coulomb field leads to theenergy eigenvalue spectrum shown schematically in Fig. 6. The conventional spectro-scopic classification of the levels in the non-relativistic limit is indicated alongside thelevels.

When the spectrum is unscrambled in this fashion, it is clear that the arrangement ofthe pair of levels for a fixed value of the total spinJ resembles a ‘supersymmetric’ pairingin which the ‘fermionic’ ladder has a spectrum identical to the ‘bosonic’ ladder exceptfor the missing ground state. The level scheme in Fig. 6 corresponds to a juxtapositionof one such ‘supersymmetric’ pair of ladders for each possible value ofJ. We now showthat the eigenvalue spectrum and the eigenfunctions of the Dirac equation may indeedbe obtained using the methods of SUSYQM.

Adopting the notation used in Bjorken and Drell [32] and defining the parameters

γ =ze2

ch, α1 = m+E , α2 = m−E , (122)

FIGURE 6. Schematic eigenvalue spectrum of the Dirac equation for the central Coulomb potentialV(r) =−γ/r. J is the total spin andk =±(J+1/2). The quantum numbers are explained in the text.

the coupled radial equations satisfied by the two-component eigenfunction(Gk,Fk) maybe written in the matrix form

(dGk/dr 0

0 dFk/dr

)+

1r

(k −γ−γ k

)(GkFk

)=

(0 α1

α2 0

)(GkFk

), (123)

in which k is an eigenvalue of the operator−(σ · L + 1) with the allowed valuesk =±1,±2,±3, . . . and satisfies|k|= J+1/2. In the representation in which Eq. (123)is written,J andk are good quantum numbers.Gk is the ‘large’ component in the non-relativistic limit. The radial functionsGk andFk must be multiplied by appropriate two-component angular eigenfunctions to make up the full four-component solutions of theDirac equation. Fig. 6 shows that when we compare the two ladders of levels for a fixedvalue ofJ, a pair of degenerate levels corresponds to the same value ofJ but oppositevalues ofk except for the lowest state of the pair of ladders when only the negative valueof k corresponds to an eigenstate. We now show how this ladder structure may be relatedto the pairing of states characteristic of supersymmetric theories (Sukumar [33]).

The matrix multiplying1/r in Eq. (123) may be diagonalized using the matrixD givenby

D =(

k+s −γ−γ k+s

), s= (k2− γ2)1/2 , (124)

and its inverse. Mutiplication of the matrix differential equation from the left byD andintroduction of the new variableρ = Er leads to

(ks+

mE

)F =

(+

ddρ

+sρ− γ

s

)G ,

(ks−m

E

)G =

(− d

dρ+

sρ− γ

s

)F . (125)

where (GF

)= D

(GkFk

). (126)

These equations are similar to the relation between the two components of the eigen-functions of a supersymmetric Hamiltonian

H = {Q , Q†} , Q =(

0 0A−0 0

), Q† =

(0 A+

00 0

), (127)

and

A±0 =(± d

dρ+

sρ− γ

s

). (128)

The nilpotent operatorQ commutes withH and therefore corresponds to a conservedcharge of this system.Q andQ† induce transformations between the ‘bosonic’ sectorrepresented byF and the ‘fermionic’ sector represented byG. Eq. (125) may be viewedas a representation of such a transformation. In supersymmetric Quantum Mechanics the‘fermionic’ and ‘bosonic’ components have identical spectra except for the ground stateof the ‘bosonic’ sector which is annihilated by the charge operatorQ.

In other words the eigenvalue equations forF andG

A+0 A−0 F =

(γ2

s2 +1−m2

E2

)F ,

A−0 A+0 G =

(γ2

s2 +1−m2

E2

)G , (129)

show that every eigenvalue ofA+0 A−0 is also an eigenvalue ofA−0 A+

0 except whenA−0 F =0. The conditionA−0 F = 0 leads to the ground state eigenfunction

F(0) = ρs exp(−γρ/s) , (130)

and the ground state energy eigenvalue is given by

(E(0)

F

)2= m2/(1+ γ2/s2) . (131)

A−0 A+0 has no normalizable eigenstate at this energy. All the other states ofA+

0 A−0 andA−0 A+

0 are paired and the eigenfunctions are linked in the formF ∼ A+0 G andG∼ A−0 F

as indicated in Eq. (125). We now show how the ground state ofA−0 A+0 may be obtained.

The uncoupled second order differential equations forF andG

[d2

dρ2 +2γρ− s(s−1)

ρ2 +1−m2

E2

]F = 0 , (132)

[d2

dρ2 +2γρ− s(s+1)

ρ2 +1−m2

E2

]G = 0 , (133)

show that Eq. (133) may be obtained from Eq. (132) by the replacements→ s+1. Thissuggests that Eq. (133) may be written in the form

A+1 A−1 G =

[γ2

(s+1)2 +1−m2

E2

]G , (134)

with

A±1 =[± d

dρ+

s+1ρ

− γs+1

]. (135)

G has a supersymmetric partnerH which satisfies

A−1 A+1 H =

[γ2

(s+1)2 +1−m2

E2

]H . (136)

Just asF andG may be viewed as the component eigenfunctions of a supersymmetricHamiltonian so alsoG andH may be viewed as the component eigenfunctions of anothersupersymmetric Hamiltonian. The spectrum ofH is identical with that ofG except fora missing state at the energy corresponding to the ground state eigenvalue ofA+

1 A−1 . By

the same reasoning as forA+0 A−0 we may then infer that the ground state ofA+

1 A−1 hasthe eigenfunction

G(0) = exp[−γρ/(s+1)] , (137)

with a ground state energy eigenvalue which satisfies

(E(0)

G

)2=

m2

1+ γ2/(s+1)2 . (138)

The excited states ofA+1 A−1 satisfy equations similar to Eq. (125). Explicitly

[k

s+1+

mE

]G = A+

1 H ,

[k

s+1−m

E

]H = A−1 G . (139)

Having obtained the ground stateG(0) we can now use the supersymmetric pairing ofFandG to obtain the first excited state ofA+

0 A−0 in the form

F(1) ∼ A+0 G(0) , E(1)

F = E(0)G , (140)

where the suffixes carry obvious meaning.This procedure may be repeated to find the ground state of a hierarchy of operators

A+2 A−2 , A+

3 A−3 , . . . with each iteration corresponding to a shift ofsby 1 in the definition oftheA± operators. This hierarchy corresponds to the Hamiltonian hierarchy discussed in§1. From the ground state properties of the members of this hierarchy all the excitedstate eigenfunctions and eigenvalues ofA+

0 A−0 can be obtained. The allowed energyeigenvalues of Eq. (125) are

E(n)F =

m√1+ γ2/(s+n)2

, n = 0,1,2, . . . , (141)

and the eigenfunctions may be written in the form

F(n) ∼ (A+

0 A+1 . . .A+

n−1

)ρs+nexp[−γρ/(s+n)] , (142)

with

A±n =± ddρ

+s+n

ρ− γ

s+n. (143)

AlthoughF andG satisfy uncoupled second order differential equations, the normal-ization of one of them determines that of the other as is required by Eq. (125). Thesolution of Eq. (123) can be written, after adding the suffixn, in the form

(G(n)

k

F(n)k

)=

12s(k+s)

(k+s γ

γ k+s

)((k/s−m/E)−1A−0 F(n)

F(n)

). (144)

When n = 0, G has no normalizable eigenstate at energyE(0)F . We treat the two

possible values ofk for this energy separately. Whenk = −sm/E(0)F Eq. (125) requires

that we must choose

F(−k,ρ) = ρsexp(−γρ/s) , G(−k,ρ) = 0 , (145)

to obtain normalizable solutions forFk andGk. However, whenk = + sm/E(0)F Eq. (125)

requires that we must choose

F(+k,ρ) = ρsexp(−γρ/s) ,

G(+k,ρ) = ρ−sexp(+γρ/s)∫ ρ

x2sexp(−2γx/s)dx . (146)

Since G is not normalizable, the positive value ofk does not lead to normalizablesolutions forFk andGk.

The expression for the spectrum, Eq. (141), is an even function ofk. A fixed |k| leadsto a doublet of states corresponding tok= +|k| andk=−|k| degenerate in energy for allpositive integral values ofn. Forn= 0, only the negative value ofk leads to normalizableFk andGk and therefore the state withn= 0 is a singlet. This explains the ladder structureof the spectrum for a fixed value of|k|= J+1/2 and opposite values ofk.

The above analysis considered a fixedk with the correspondings = (k2− γ2)1/2.However, in Eqs. (141)-(143)s enters only as a parameter. Hence we can obtain thecomplete spectrum and the eigenfunctions of Eq. (123) for all values ofJ = |k|−1/2 bythe above procedure. The complete spectrum is given by

E(n)k =

m√1+ γ2/

[n+(k2− γ2)1/2

]2, n = 0,1,2, . . . , k = 1,2, . . . , (147)

and Eq. (142) for the eigenfunctionsF(n) is valid for different values ofs(k). Theprincipal quantum numberN is related ton andk by the relationN = n+ |k|.

We have shown that the supersymmetric pairing ofF andG enables an elegant treat-ment of the Dirac equation for a central Coulomb field (Darwin [34], Biedenharn [35]).The interaction of a charged particle with the vacuum fluctuations of the quantized radi-ation field leads to departures from the Coulomb potential. When the field deviates froma Coulomb field, the transformed statesF andG no longer belong to a supersymmetricpair and the eigenvalue spectrum loses the ladder structure characteristic of supersym-metric pairing. Therefore we may say that the Lamb shift, which is an effect due tovacuum fluctuations, is related to the breaking of the supersymmetry that connectsFandG.

4. SUPERSYMMETRY, POTENTIALS WITH BOUND STATES ATARBITRARY ENERGIES AND MULTI-SOLITON SOLUTIONS

OF KDV EQUATION

The connection between the algebra of supersymmetry and the inverse scattering methodcan be used to construct reflectionless potentials with any specified number of non-degenerate bound states at arbitrary energies. The reflection coefficient of the potentialso constructed is related to the reflection coefficient of a potential which supports nobound states. By choosing the reference potential to beV = 0 it is possible to constructreflectionless potentials with bound states at arbitrary energies. It is well known thatthe KdV equation has muti-soliton solutions with known analytical form. TheN-solitonand the (N + 1)-soliton solutions of the KdV equation may be shown to be connectedby a transformation identical to that of the transformation linking the eigenfunctions ofthe bosonic and fermionic sectors of a Supersymmetric pair. The reflectionless potentialwith N bound states constructed using the ideas of the SUSY method to plant boundstates can be shown to be identical to theN-soliton solution of the KdV equation forappropriate choice of a set of parameters.

4.1. Potentials with a single bound state

Let V0(x) be a potential that supports no bound states andR0(k) be the reflectioncoefficient for positive energies. Using the procedure outlined in §1.7 ([18]) it is possibleto find a potentialV1 which supports a single bound state at energyE1 =−γ1

2/2. It wasshown in §1.7 thatV1 may be written in the form

V1 = V0− d2

dx2 lnψ0(E1) , (148)

whereψ0(E1) is a nodeless unnormalizable solution of the Schrödinger equation for thepotentialV0 at energyE1. It was shown in §1.7 that the ground state eigenfunction ofV1at energyE1 is given by

ψ1(E1)∼ 1ψ0(E1)

, (149)

while for E 6= E1ψ1(E)∼ A−0 (E1)ψ0(E) , (150)

where

A−0 (E1) =1√2

[− d

dx+

ddx

lnψ0(E1)]

. (151)

The reflection coefficient ofV1 is given by

R1(k) =γ1− ikγ1 + ik

R0(k) . (152)

The above results simplify for the case of a free particle for whichV0 = 0 andR0(k) = 0.Eq. (152) shows that the reflection coefficient for the supersymmetric partner vanishesidentically. The non-normalizable solution inV0 for energyE1 is given by

ψ0(E1) = coshγ1x+α1sinhγ1x , |α1|< 1 . (153)

The condition|α1| < 1 ensures thatψ0(E1) is a nodeless function even though it is notnormalizable. The reflectionless potentialV1 with a single bound state atE1 is given by

V1 =−γ21sech2

(γ1x+ tanh−1α1

), (154)

with the ground state eigenfunction

ψ1(E1)∼ sech(γ1x+ tanh−1α1

). (155)

Whenα1 = 0, V1 is a symmetric reflectionless potential. Using the suffix SR to denote‘symmetric and reflectionless’ and using the notation that the normalized eigenfunctionswill be denoted by the addition of a tilde, in terms of the normalized ground stateeigenfunction

ψ1SR(E1) = (γ1/2)1/2sechγ1x , (156)

the potentialV1SRmay be written in the form

V1SR=−2γ1ψ21SR(E1) . (157)

4.2. Potentials with two bound states

The procedure used in the previous subsection may be repeated to find a potentialwith two bound states at energiesE1 andE2. V2 is given by

V2 = V1− d2

dx2 lnψ1(E2) , (158)

whereψ1(E2) is the nodeless non-normalizable solution of the Schrödinger equation forthe potentialV1 at energyE2 =−γ2

2/2. The ground state eigenfunction ofV2 is given by

ψ2(E2)∼ 1ψ1(E2)

, (159)

while the eigenfunctions for other energies are given by

ψ2(E)∼ A−1 (E2)ψ1(E) , E 6= E2 , (160)

where

A−1 (E2) =1√2

[− d

dx+

ddx

lnψ1(E2)]

. (161)

In particular the first excited state ofV2 at energyE1 has the eigenfunction

ψ2(E1)∼ A−1 (E2)ψ1(E1) . (162)

Using Eqs. (148) and (158) the potentialV2 may be written in the form

V2 = V0− d2

dx2 ln [ψ0(E1)ψ1(E2)] . (163)

The reflection coefficient ofV2 for positive energies is given by

R2(k) =(γ2− ik)(γ1− ik)(γ2 + ik)(γ1 + ik)

R0(k) . (164)

The above expressions forV2 andψ2 are given in terms of the solutionψ1 in the potentialV1. It would be more convenient to express all quantities in terms of the solutions in thereference potentialV0 which has no bound states. Eqs. (150) and (151) show that

ψ1(E2)∼[− d

dx+

ddx

lnψ0(E1)]

ψ0(E2) . (165)

Henceψ0(E1)ψ1(E2)∼ detD2 , (166)

whereD2 is a matrix given by

D2 =(

ψ0(E1) ψ0(E2)ψ0(E1) ψ0(E2)

). (167)

These expressions may be used to write the potential with two bound states in the form

V2 = V0− d2

dx2 lndetD2 . (168)

The ground state eigenfunction is given by

ψ2(E2)∼ ψ0(E1)detD2

∼ [D−1

2

]22 . (169)

The eigenfunction for the first excited state ofV2 may be simplified to the form

ψ2(E1)∼ ψ0(E2)detD2

∼ [D−1

2

]12 . (170)

Thus the potentialV2 is expressed in terms of the second derivative of the determinant ofD2 while the eigenfunctions ofV2 are given in terms of the elements in the last columnof the inverse of the matrixD2. The condition thatψ0(E1) andψ1(E2) must be chosento be nodeless is equivalent to the requirement thatψ0(E1) andψ0(E2) must be chosensuch that the determinant ofD2 is free of zeros.

To illustrate the above results we consider the case of the free particle,V0 = 0. For thiscase the reflection coefficient for the potential with two bound states vanishes. Since

ψ0(E1) = coshγ1x+α1sinhγ1x ,

ψ0(E2) = sinhγ2x+α2coshγ2x , (171)

the condition that the determinant ofD2 be free of zeros can be met only if|α1|< 1 and|α2|< 1. The symmetric reflectionless potential with bound states atE1 andE2, obtainedby choosingα1 = 0 andα2 = 0, is given by

V2SR=− d2

dx2 lndetD2SR , (172)

where

D2SR=(

coshγ1x sinhγ2xγ1sinhγ1x γ2coshγ2x

). (173)

The potential may be reduced to the form

V2SR=−(γ22− γ2

1)γ22 cosh2γ1x+ γ2

1 sinh2γ2x

(γ2coshγ2xcoshγ1x− γ1sinhγ2xsinhγ1x)2 . (174)

The normalized eigenfunctions ofV2SRmay be written in the form

ψ2SR(E2) =(γ2

2

(γ22− γ2

1

))1/2 coshγ1xdetD2SR

,

ψ2SR(E1) =(γ1

2

(γ22− γ2

1

))1/2 sinhγ2xdetD2SR

. (175)

In terms of these normalized eigenfunctions the symmetric reflectionless potential maybe written in the form [36]

V2SR=−2[γ2ψ2

2SR(E2)+ γ1ψ22SR(E1)

]. (176)

Fig. 7 shows some examples of symmetric reflectionless potentials with two bound statesfor certain choices of bound state energies.

Certain features ofV2SRmay be analytically established:(i) if E2 > 3E1 thenx = 0 is a minimum of the potential and there are no additional

minima andV2SRis a single well. In particular ifE2 = 4E1 the resulting potential has thesimple form

V2SR=−3γ21sech2γ1x , (177)

i.e., V2SR is a sech2 potential with bound states at−2γ21 and−γ2

1/2.(ii) if E2 < 3E1 then x = 0 is a maximum of the potential and there is a pair of

additional minima for|x| 6= 0 andV2SR under these conditions is a symmetric doublewell. These features are illustrated in Fig. 7.

FIGURE 7. Symmetric reflectionless potentials with bound states at energiesE1 = −γ12/2 andE2 =

−γ22/2 for γ1 = 1 andγ2 = (a)1.1, (b) 1.3, (c) 1.5, (d) 2.0. The locations of the bound levels are indicated

by broken lines.

4.3. Potentials with an arbitrary number of bound states

By an extension of the procedure outlined in the last two sections it is possible toconstruct a hierarchy of Hamiltonians with successively increasing number of boundstates starting from the HamiltonianH0 with no bound states [21, 36]. Denoting theHamiltonian withn bound states byHn and the ground state energy ofHn by En:

En =−γ2n/2 , γ2

n > γ2n−1 > .. . > γ2

1 , (178)

the Hamiltonian hierarchy is given by

Hm = A−m−1(Em)A+m−1(Em)+Em

= Hm−1 +[A−m−1(Em),A+

m−1(Em)]

, m= 1,2, . . . ,n , (179)

where

A±m−1(Em) =1√2

[± d

dx+

ddx

lnψm−1(Em)]

, (180)

andψm−1(Em) is a non-normalizable nodeless solution of the eigenvalue equation forHm−1 at energyEm which lies below the ground state ofHm−1. The potentials in the

hierarchy are related by

Vm = Vm−1− d2

dx2 lnψm−1(Em) . (181)

The ground state eigenfunction ofHm is given by

ψm(Em)∼ 1ψm−1(Em)

, (182)

while all the other eigenfunctions ofHm are given in terms of the eigenfunctions ofHm−1by

ψm(Ei)∼ A−m−1(Em)ψm−1(Ei) , i = 1,2, . . . ,m−1 , m= 1,2, . . . ,n . (183)

This network of interrelated eigenfunctions can be disentangled to expresss all eigen-functions of Hm in terms of the solutions in the reference potentialV0. Iteration ofEq. (181) shows that the potential withn bound states is related toV0 by

Vn = V0− d2

dx2 [lnψ0(E1)ψ1(E2) . . .ψn−1(En)] . (184)

It is possible to express the product of eigenfunctions in the above equation in termsof the solutionsψ0(Ei) in the potentialV0 for various energiesEi . It is then possible toexpressVn in the form

Vn = V0− d2

dx2 lndetDn , (185)

where the matrixDn is given by

[Dn] jk =d j−1

dxj−1ψ0(Ek) , j,k = 1,2, . . . ,n . (186)

The eigenfunctions for the potentialVn may be expressed in the form

ψn(Ei)∼[D−1

n

]in , i = 1,2, . . . ,n . (187)

The proof that the elements in the last column of the inverse of the matrixDn are indeedthe eigenfunctions forVn in Eq. (185) with eigenenergiesEi , i = 1,2, . . . ,n, is straight-forward but involves a long algebraic calculation. The requirement thatψm−1(Em),m = 1,2, . . . ,n, be nodeless can be met by choosing the non-normalizable solutionsψ0(Em),m = 1,2, . . . ,n, such that the determinant ofDn has no zeros. The eigenfunc-tion relation given in Eq. (183) can be extended to positive energies to show that thereflection coefficient ofVm is related to the reflection coefficient ofVm−1 by

Rm(k) =γm− ikγm+ ik

Rm−1(k) . (188)

Iteration of this relation gives

Rm(k) =

[∏

m=1,2,...,n

(γm− ikγm+ ik

)]R0(k) . (189)

Eqs. (185)-(187) provide a recipe for constructing potentials with bound states at speci-fied energiesEm and reflection coefficient for positive energies given by Eq. (189).

The algorithm for constructing reflectionless potentials withn bound states is a par-ticular case of the procedure given above corresponding to the choiceV0 = 0. SinceR0(k) = 0 whenV0 = 0, Rn(k) also vanishes. The free particle solutions at energiesE jare given by

ψ0(E j) =12

[exp

(γ jx+θ j

)+(−) j+1exp

(−γ jx−θ j)]

, (190)

whereθ j are arbitrary phase factors. For odd values ofj, ψ0(E j) is acoshfunction andhence nodeless while for even values ofj, ψ0(E j) is asinhfunction with a single node.Such a choice of solutions ensures that detDn has no zeros. A symmetric reflectionlesspotential withn bound states may be obtained by choosingθ j = 0 for all values of j.The potential so obtained is given by

VnSR=− d2

dx2 lndetDnSR, (191)

where the elements of the matrixDnSRare given by

[DnSR] jk =(γk)

j−1

2

[exp(γkx)+(−1) j+k exp(−γkx)

]. (192)

The unnormalized eigenfunctions of this potential may be given in terms of the elementsin the columnn of the inverse of the matrixDnSRand the normalized eigenfunctions maybe written in the form

ψnSR(Ei) =

(γi

2

n

∏k6=i

|γ2k − γ2

i |)1/2[

D−1nSR

]in , i = 1,2, . . . ,n . (193)

The relationship of this representation of the symmetric reflectionless potentials to otherapparently different representations of the same potential can be established.

It can be shown [36] that the symmetric reflectionless potential withn bound statesmay be represented in terms of the normalized bound state eigenfunctions in the form

VnSR=−2n

∑j=1

[γ jψ2

nSR(E j)]

, (194)

which is a generalized form of the results given in Eqs. (157) and (176) for the casesn = 1 andn = 2.

The analysis of Kay and Moses [37] and then-soliton solution of the Korteweg-deVries equation [38, 39, 40] lead to the result that the symmetric reflectionless potentialmay be expressed in terms of a matrixM with elements

M jk = δ jk +λ j(x)λk(x)

γ j + γk, (195)

whereδ jk is the Kronecker delta function and

λ j(x) = Cj exp(−γ jx) ,C2

j

2γ j= ∏

k6= j

(γk + γ j

γk− γ j

), (196)

in the form

V =− d2

dx2 lndetM . (197)

It can be shown that the matricesM andDnSRare related by

M = 2GA−1DnSRG−1 , (198)

where the elements ofG andA are given by

Gi j = δi j

(γ j

2 ∏k6= j

∣∣γ2k − γ2

j

∣∣)1/2

,

Ai j = γ i−1j exp(γ jx) . (199)

It is clear that

detA ∝ exp

(∑i

γix

). (200)

Using Eqs. (198)-(200) it is easy to show that

d2

dx2 lndetM =d2

dx2 lndetDnSR, (201)

leading to the result that the instantaneousn-soliton solution of the KdV equation givenby Eqs. (195)-(197) is identical to the symmetric reflectionless potential constructedusing the methods of Supersymmetric Quantum Mechanics. These ideas will be furtherexplored in the next subsection.

4.4. Backlund transformation, KdV equation and supersymmetry

A potential of the form given in Eq. (154) can be regarded as the instantaneous one-soliton solutionV1(x) = v1(x,0) of the Korteweg-deVries equation [38] in the form

(∂ 3

∂x3 −12v∂∂x

+∂∂ t

)v(x, t) = 0 , (202)

with v = v1. The solution for allt is

v1(x, t) =−γ21sech2

(γ1x−4γ3

1t− tanh−1α1)

. (203)

and the soliton travels with a velocity proportional to its amplitude. Then-solitonsolution of the KdV equation (202) has been studied extensively [37, 39, 40]. The timedependentn-soliton solution may be given in the form of Eqs. (195)-(197) with thechoice of

λ j(x) = Cj exp(−γ jx+4γ3

j t)

, (204)

by endowing the basis functions with a time dependence of a specific form. It is thisspecific time dependence that guarantees that then-soliton solution satisfies the partialdifferential Eq. (202). Similarly, a time dependence for the same potential constuctedusing SUSY and given by Eqs. (184)-(186) and (190) can be introduced by allowing thephasesθ j to have a time dependence of the form

θ j(t) = θ j(0)−4γ3j t . (205)

Such a choice of time dependence guarantees that the potentialVn satisfies the KdVequation (202). It is possible to verify these assertions by a long calculation.

It is possible to consider other choices of time dependence for the phases. It can beshown that if the phases are allowed to vary with time in the form

θ j(t) = θ j(0)−2m−1γmj t , (206)

where the indexm can take all odd values≥ 3 then each of these cases can lead to asolution of a member of the Kadomtsev-Petviashvili (KP) hierarchy studied by Caudreyet al. [41]. m = 3 leads to the KdV equation (202) whilem = 5,7, . . . lead to highermembers of the KP hierarchy (Sukumar [42]). For this class of non-linear equations then-soliton solution is given by Eqs. (185), (186), (190) and (206). For examplem = 5leads to the non-linear equation

(∂ 5

∂x5 −20V∂ 3

∂x3 −40∂V∂x

∂ 2

∂x2 +120V2 ∂∂x

+∂∂ t

)V(x, t) = 0 , (207)

which has been studied by Sawada and Kotera [43] and Caudreyet al [41].In the context of the KdV equation the transformations of solutions possessingn−1

solitons to those withn solitons are known as Backlund transformations [44]. TheBacklund transformation for the KdV equation can be understood as follows. If weconsider a functionF(x, t) satisfying a modified KdV equation of the form

G(x, t)≡ ∂F∂ t

+12(γ2−F2) ∂F

∂x+

∂ 3F∂x3 = 0 , (208)

then it can be established that the functionv(x, t) defined by

v(x, t)≡ 12

(F2− ∂F

∂x− γ2

), (209)

can be shown to satisfy the KdV equation since Eq. (202) is equivalent to the equation

2FG− ∂G∂x

= 0 , (210)

which is clearly satisfied becauseF satisfies Eq. (208). Also it is evident that by similarreasoning

2FG+∂G∂x

= 0 , (211)

which implies that the functionv(x, t) is defined by

v(x, t)≡ 12

(F2 +

∂F∂x

− γ2)

, (212)

also satisfies the KdV equation (202). With suitable boundary conditions onF , one caninterpretv as ann-soliton solution ifv is an(n−1)-soliton solution. By eliminatingFthe relation betweenv andv may be established to be

∂∂x

[(v+ v+ γ2)1/2

]= v−v . (213)

We now show that this relation betweenv andv is consistent with the relation betweensupersymmetric partner potentials defined by Eqs. (178)-(182). If we define the function

F(x) =ddx

lnψm(Em) . (214)

then using Eqs. (178)-(182) it is easy to show that the potentialsVn andVn−1 can bewritten in terms ofF in the form

Vn =12

(F2 +

ddx

F− γ2n

),

Vn−1 =12

(F2− d

dxF− γ2

n

). (215)

The above equations are identical to Eqs. (209) and (212) which definev andv. Thus Su-persymmetric Quantum Mechanics enables the identification of the functionF definedby the Backlund transformation (Eq. (209)) for the KdV equation as the logarithmicderivative of the ground state eigenfunction of the potentialVn(x, t) which supportsnbound states where the time dependence arises from the time dependence of theθ j inEqs. (190) and (205). The above argument can be extended to interpret the Backlundtransformation for the entire system of non-linear equations defined by the KP hierar-chy.

4.5. Summary

We have shown that by repeatedly using the algebra of supersymmetry in a stepby step fashion it is possible to construct potentials with bound states at arbitrary

energies. It has been shown that the non-normalizable solutions in a reference potentialwhich supports no bound states constitute the input in this construction. The reflectioncoefficient of the potential withn bound states constructed by this procedure is relatedto the reflection coefficient in the reference potential.Vn is in general not only a functionof the n bound state energiesEi but also a function ofn parametersθi , i = 1,2, . . . ,n.θi characterizes a particular linear superposition of the two linearly independent non-normalizable solutions in the reference potentialV0 at energyEi . θi can take such valuesthat ensure that the determinant ofDn is free of zeros.

By choosing the reference potential to beV0 = 0 reflectionless potentials withn boundstates may be constructed. The reflectionless potential so obtained is not necessarily asymmetric function ofx. By choosing the parametersθi to have specific values sym-metric reflectionless potentials can be constructed. The resulting symmetric potential isidentical to the one constructed from then-soliton solution of the KdV equation. Sym-metric reflectionless potentials are unique and interesting because they are specified en-tirely by their bound states. They have been shown to provide good approximations toconfining potentials, such as those which confine quarks, in the range of energies actu-ally probed by the levels (Thackeret al.[45], Quigg and Rosner [46], Kwong and Rosner[47]). Kwong and Rosner have constructed an approximation to a quarkonium potentialon the basis of information about itsnS levels. Using the energies and leptonic widthsof 10 levels below the flavour threshold Kwong and Rosner have used the algorithm forconstructing symmetric reflectionless potentials to construct a potentialV(r). Calcula-tions such as these provide information on the distance scale over whichtt bound stateinformation is likely to shed light on the interquark force.

We have shown that it is possible to identify a hierarchy of non-linear equations ofthe KdV type which have the property that then-soliton solution of these equations canbe explicitly given in analytic form. The connection between SUSYQM and Backlundtransformations provides interesting insights and enables a physical interpretation of thefunction that is used to construct the Backlund transformation. The symmetries whichlead to an infinite number of conservation laws in integrable systems such as the KdVequation have already been recognized as being related to Kac-Moody algebras. It isclear that we can now add supersymmetry to the list of properties connected with suchsystems.

We have also shown that the symmetric reflectionless potentials may be expressed interms of the normalized bound state eigenfunctions in a particularly simple form. It isclear that the construction based on supersymmetry not only agrees with calculationsbased on other procedures but also provides unique insight into the structure of symmet-ric reflectionless potentials and the structure of the multi-soliton solutions of the KdVhierarchy and the Backlund transformations associated with these non-linear equations.

5. PAIRS OF SUSY TRANSFORMATIONS FOR THE RADIALSCHRÖDINGER EQUATION

It was shown in an earlier lecture that it is possible to identify four different transforma-tions by which one can find a supersymmetric partner to a given radial Schrödinger equa-

tion. The modifications of the Jost function and the singularity structure in the asymp-totic regionr → ∞ and in the regionr → 0 for the 4 types of SUSY transformationswere identified. Each of the transformations alters the spectral density in a specific man-ner and either removes or adds a bound state or maintains the same spectrum. In thislecture we study how pairs of SUSY transformations may be used to modify or maintainthe spectrum with or without altering the spectral density. An exactly solvable exampleis used to illustrate the procedure.

5.1. Classification of the four types of SUSY transformations

In §2 four different transformations of the radial Schrödinger equation were identifiedby T1, T2, T3 andT4. The potentials, eigenstates and phase shifts and the Jost functionafter the supersymmetric transformation were denoted by adding a tilde. The differenttypes of transformations were distinguished by adding a suffix. In this lecture we adoptthe same notations for single transformations. Succesive transformations will be indi-cated by adding further suffixes and tildes. We also use the same notation as in §2 thatthe regular solution at a given energyE = −γ2/2 will be denoted byφ(r) and the Jostsolution byη(r). It was shown that these functions obey the boundary conditions

limr→0

φ(r) =r(l+1)

(2l +1)!!, lim

r→∞φ(r)∼ exp(γr) , (216)

limr→0

η(r)∼ r−l , limr→∞

η(r)∼ exp(−γr) . (217)

We also use the notationψ(m)(r) is an abbreviation forψ(r,E(m)).It was shown in §2 that for all the four types of transformations the potential is

unaltered in the regionr →∞ and that each of the transformations produces an alterationof the potential in the region of small and mediumr and nearr → 0 the alteration ofthe potential is equivalent to altering the angular momentuml in the centrifugal partl(l + 1)/r2 of the potentialV(r) corresponding to a definite partial wave. The changesfor the four transformations corresponding to the factorization energyE =−γ2/2 wereidentified as:

T1 : F = Fk

k− iγ, limr→0 l → l +1 , δ = δ − tan−1(γ/k)− π

2,

T2 : F = Fk− iγ

k, limr→0 l → l −1 , δ = δ + tan−1(γ/k)+

π2

,

T3 : F = Fk

k+ iγ, limr→0 l → l +1 , δ = δ + tan−1(γ/k)− π

2,

T4 : F = Fk+ iγ

k, limr→0 l → l −1 , δ = δ − tan−1(γ/k)+

π2

. (218)

T1 eliminates the ground state atE leaving the rest of the spectrum of eigenvaluesunaltered.T2 adds a new ground state atE while leaving the rest of the spectrumunaltered.T3 and T4 maintain the same spectrum and alter the Jost function and the

singularity of the potential at the origin. These four transformations may be viewedas the building blocks which can be used in suitable combinations to produce desiredmodifications of the spectrum, the Jost function and the singularity at the origin. Wenext study pairs of SUSY transformations and the alteration of the spectrum and the Jostfunction that they produce.

5.2. Pairs of SUSY transformations

We consider two successive transformations performed at the same energyE. Of thesixteen possibilities that arise from a combination of any one of theTj with any otherTk, it is clear whenj andk are different the combinationTjTk is equivalent toTkTj . Ofthe remaining ten possibilitiesT1T1 does not exist because it is not possible to removethe same bound state twice andT2T2 does not exist because the same bound state cannot be added twice. Therefore there are eight distinct possibilities for pairs of SUSYtransformations which must be considered. We now classify them.

(1) T1 followed by T2 removes a bound state at energyE and adds the same boundstate at the same energy. At the end of the two transformations we get a potential whichhas the same spectrum as the original potential but the normalization constant of theground state may be altered and the Jost function is unaltered.

˜F12 = F . (219)

The phase shifts and the normalization constants of all the excited state eigenfunctionsare unaltered. ThereforeT2T1 produces the phase-equivalent family corresponding to agiven potential.

(2) T1 followed by T3 removes a bound state at energyE and alters the spectraldensity and the normalization constants of the remaining eigenstates. The Jost functionis modified to

˜F13 = Fk2

k2 + γ2 . (220)

The phase shifts for positive energies are unaltered. However, the resulting potential issingular with a singularity of the formr−2.

(3) T1 followed byT4 removes a bound state at energyE and keeps the spectral densityunaltered which implies that the normalization constants of all the other bound states areunaltered. The Jost function is modified to

˜F14 = Fk+ iγk− iγ

. (221)

The phase shifts for positive energies will be altered.(4) T3 followed by T2 adds a bound state below the ground state of the original

potential without altering the spectral density. The normalization constants of all theother states are left unaltered. The Jost function is modified to

˜F32 = Fk− iγk+ iγ

. (222)

The phase shifts for positive energies will, however, be altered.(5) T2 followed by T4 adds a bound state below the ground state of the original

potential and alters the spectral density and the normalization constants of the othereigenstates. The Jost function is modified to

˜F24 = Fk2 + γ2

k2 . (223)

The phase shifts for positive energies will not be altered. HenceT4T2 produces a familyof phase-equivalent potential with a new ground state and different members of thefamily will have different values for the normalization constants of the ground state.The resulting potential is singular at the origin.

(6) T3 followed by anotherT3 maintains the same spectrum and alters the spectral den-sity. The normalization constants of all the states are altered and the resulting potentialhasr−2 singularity at the origin. The Jost function is modified to

˜F33 = F

(k

k+ iγ

)2

. (224)

The phase shifts for positive energies and the normalization constants of the eigenfunc-tions will be altered.

(7) T3 followed by T4 transforms the original Hamiltonian back to itself withoutaltering the spectral density, spectrum and the phase shifts for positive energies. In fact

˜F34 = F . (225)

(8) T4 followed by anotherT4 is similar to case (6) and also maintains the samespectrum, alters the spectral density and the normalization constants of all the states.The resulting potential is a singular potential. The Jost function is modified to

˜F44 = F

(k+ iγ

k

)2

. (226)

The phase shifts for positive energies and the normalization constants of the eigenfunc-tions will be altered.

The eight cases listed above exhaust the possible combinations of two SUSY trans-formations. We now examine these cases in detail.

(1) We now show thatT1 followed by T2 generates the phase-equivalent family to agiven potential

V(r) =l(l +1)

2r2 +v(r) , (227)

with ground stateψ(0)(r) at energyE(0). After the first transformationT1 which elimi-nates the ground state ofV the potential is

V1(r) = V(r)− d2

dr2 lnψ(0)(r) . (228)

Using the second transformationT2 the eliminated state can be introduced as the newground state below the ground state ofV1. It was shown in §2 that the solution inV1 atthe correspoding energy may be given in the form

ψ1(r,E(0),α) = φ1(r,E(0))cosα + η1(r,E(0))sinα , (229)

whereφ1 is the regular solution inV1 at energyE(0) andη1 is the Jost solution inV1 atthe same energy and are given by

φ1 ∼ 1

ψ(0)(r)

∫ r

0

(ψ(0)(x)

)2dx ,

η1 ∼ 1

ψ(0)(r)

∫ ∞

r

(ψ(0)(x)

)2dx . (230)

ψ can be written in terms of the parameterλ defined by

tanα =1

λ +1, (231)

in the form

ψ1(r,E(0),λ ) =1+λ

∫ r0

(ψ(0)(x)

)2dx

ψ(0)(r), −1 < λ < ∞ . (232)

The potential at the end of the two transformations is given by

˜V1,2(r) = V(r)− d2

dr2 ln

(1+λ

∫ r

0

(ψ(0)(x)

)2dx

), (233)

which has a spectrum identical with that ofV(r) and has normalized ground stateobtained from the inverse ofψ in Eq. (232) in the form

˜ψ(0)1,2 =

(1+λ )1/2ψ(0)(r)

1+λ∫ r

0

(ψ(0)(x)

)2dx

. (234)

The excited states at the end of the two transformations are given by

˜ψ(m)1,2 =−

(E(m)−E(0)

)−1A−2 A−1 ψ(m) , m= 1,2, . . . , (235)

where

A−2 (E(0),λ ) =1√2

[− d

dr+

ddr

ln ψ1

],

A−1 =1√2

[− d

dr+

ddr

lnψ(0)(r)]

. (236)

The excited state eigenfunctions can be simplified to the form

˜ψ(m)1,2 = ψ(m)−λ

ζ (m)(r)2(E(m)−E(0)

) ,

ζ (m)(r) =ψ(0)(r)

1+λ∫ r

0

(ψ(0)(x)

)2dx

(ψ(m) d

drψ(0)−ψ(0) d

drψ(m)

). (237)

The Wronskian relation arising from the the Schrödinger equation considered at twodifferent energies in the form

(ψ(m) d

drψ(0)−ψ(0) d

drψ(m)

)= 2

(E(m)−E(0)

)∫ r

0ψ(0)(x)ψ(m)(x)dx , (238)

may then be used to express the excited state eigenfunctions in the form

˜ψ(m)1,2 (r) = ψ(m)(r)−λψ(0)(r)

∫ r0 ψ(0)(x)ψ(m)(x)dx

1+λ∫ r

0

(ψ(0)(x)

)2dx

, m= 1,2, . . . . (239)

From Eqs. (234) and (239) it can be established that

˜δ 1,2(l ,k) = δ (l ,k) ,

limr→0

˜ψ(m)1,2 (r,λ ) = lim

r→0ψ(m)(r) , m= 1,2, . . . ,

limr→0

˜ψ(0)1,2(r,λ ) = (1+λ )1/2 lim

r→0ψ(0)(r) . (240)

It is also clear from Eq. (218) that at the end of the two transformationsT1 andT2 thesingularity at the origin arising from the centrifugal part of the potential is unalteredand also that the phase shifts are unaltered. These results show that the family of po-tentials ˜V1,2(r,λ ) in Eq. (233) for∞ > λ > −1 have identical spectra, identical phaseshifts and identical normalization constants for the excited states but have different nor-malization constants for the ground state for different values ofλ . Hence this familyof potentials belongs to a phase-equivalent family. These expressions for the new po-tential and the new eigenfunctions are in agreement with the results obtained using theGelfand-Levitan procedure for changing the normalization constant of the ground state[26, 30]. Thus we have shown that the Gelfand-Levitan [24] procedure for changing thenormalization constant of the ground state without changing the spectrum is equivalentto a transformation of the typeT1 followed by another suitable transformation of the typeT2 [28].

(2) We now considerT1 followed byT3. The first step is the same as case (1) and leadsto the potential given by Eq. (228). The second step corresponds to aT3 transformationwith the choice of valueα = 0 in Eq. (229) which also corresponds to the choiceλ = ∞in Eq. (232). Such a choice forψ1 leads to the new potential after the two transformationsof the form

˜V1,3(r) = V(r)− d2

dr2 ln

[∫ r

0

(ψ(0)(x)

)2dx

], (241)

which has a spectrum identical with that ofV(r) except for missing the ground state ofV. The eigenfunctions ofV1,3 are related to the excited state eigenfunctions ofV by

˜ψ(m)1,3 =−

(E(m)−E(0)

)−1A−3 A−1 ψ(m) , m= 1,2, . . . , (242)

where

A−3 =1√2

[− d

dr+

ddr

ln

(1

ψ(0)(r)

∫ r

0

(ψ(0)(x)

)2dx

)], (243)

andA−1 has the same form as in Eq. (236). From these equations it can be establishedusing the same steps as in case (1) that

˜ψ(m)1,3 = ψ(m)(r)−ψ(0)(r)

∫ r0 ψ(0)(x)ψ(m)(x)dx∫ r

0

(ψ(0)(x)

)2dx

, m= 1,2, . . . , (244)

and that the normalization constants of the eigenfunctions are altered. It can be shownfrom Eq. (241) or from Eq. (218) that

limr→0

˜V1,3(r) =(l +2)(l +3)

2r2 , limr→∞

˜V1,3(r) =l(l +1)

2r2 (245)

which shows that the new potential is singular at the origin and has a short rangerepulsive singularity. Eq. (218) can also be used to show that the phase shifts aredecreased by a constant amountπ for all positive energies which is equivalent to sayingthat the phase shifts are unaltered within moduloπ. Thus a transformation of the typeT1followed by a transformation of the typeT3 can be used to produce a singular potentialwhich has the same phase shifts as the original potential but has one less bound state[48].

(3) We next considerT1 followed byT4. The first step is the same as in cases (1) and(2) and leads to the potential given in Eq. (228). The second step corresponds to aT4tranformation with the choice of valueα = π/2 in Eq. (229) which also corresponds tothe choiceλ = −1 in Eq. (232). Such a choice ofψ1 leads to a new potential after thetwo transformations of the form

˜V1,4(r) = V(r)− d2

dr2 ln

(∫ ∞

r

(ψ(0)(x)

)2dx

), (246)

which has the same spectrum asV except for missing the ground state ofV. Theeigenstates of the new potential can be obtained by the same procedure as for the earliercases and can be readily obtained by takingλ =−1 in Eq. (239) to give

˜ψ(m)1,4 (r) = ψ(m)(r)−ψ(0)(r)

∫ ∞r ψ(0)(x)ψ(m)(x)dx∫ ∞

r

(ψ(0)(x)

)2dx

, m= 1,2, . . . . (247)

It can be established from this equation that the bound state normalizations are unaf-fected. It can be shown using Eq. (218) that the phase shift relation for positive energies

is˜δ 1,4(l ,k) = δ (l ,k)−2tan−1

(γ(0)

k

). (248)

It can also be shown using Eq. (218) that the singularity of the potential˜V1,4 at the originis unaltered. The expressions for the new potential and the new eigenfunctions [28] areidentical to the results given by Abraham and Moses [26]. Thus the Gelfand-Levitanprocedure to eliminate the ground state without introducing additional singularitiesat the origin is equivalent to a transformationT1 which eliminates the ground statefollowed by a transformationT4 which removes the additional singularity at the originwhich is introduced by the first transformation. The resultant potential after the twotransfromations is non-singular.

(4) We next considerT3 followed byT2. The transformationT3 may be used at energyE below the ground state ofV to produce a new potential which has the same spectrumas that ofV as described in §2.6. TheT3 transformation is implemented by consideringthe regular solutionφ(r, E) for the potentialV(r). After theT3 transformation the newpotential is

V3(r, E) = V(r)− d2

dr2 lnφ(r, E) . (249)

We now apply aT2 transformation at the energyE to introduce a new bound state belowthe ground state ofV. One of the solutions at energyE in the potentialV3 is given by

ξ =1

φ(r, E), (250)

and the second linearly independent solution is, therefore, given by

χ(r, E) =1

φ(r, E)

∫ r

0

(φ(x, E)

)2dx . (251)

By studying the limiting behaviour ofφ , ξ andχ it can be established that the regularand the Jost solution inV3 are indeed proportional toχ andξ respectively. Hence usingthe general solution

ψ3(r, E,α) = χ(r, E)sinα +ξ (r, E)cosα , (252)

the new potential after the two transformations may be given in the form

˜V3,2(r, E,α) = V(r)− d2

dr2 ln

(cosα +sinα

∫ r

0

(φ(x, E)

)2dx

), (253)

which has a ground state eigenvalue atE while all the other eigenvalues are identical tothose ofV. The normalized ground state of˜V3,2 is given by

˜ψ(0)3,2(r, E,α) =

(sinα cosα)1/2φ(r, E)

cosα +sinα∫ r

0

(φ(x, E)

)2dx

. (254)

Eq. (254) shows that the parameterα determines the normalization constant of theground state.α = π/4 corresponds to choosing the normalization constant as 1 andfor this choice the resulting potential is

˜V3,2(r, E,π/4) = V(r)− d2

dr2 ln

(1+

∫ r

0

(φ(x, E)

)2dx

), (255)

which has the ground state eigenfunction

˜ψ(0)3,2(r, E) =

φ(r, E)

1+∫ r

0

(φ(x, E)

)2dx

. (256)

The excited state eigenfunctions are given by

˜ψ(m)3,2 (r) =−

(E(m)−E(0)

)−1A−2 A−3 ψ(m)(r) , m= 1,2, . . . , (257)

where

A−2 =1√2

[− d

dr+

ddr

ln

(1+

∫ r0

(φ(x, E)

)2dx

φ(r, E)

)],

A−3 =1√2

[− d

dr+

ddr

lnφ(r, E)]

. (258)

Using the same method as in previous cases the expression for the excited state eigen-functions can be brought to the form

˜ψ(m)3,2 (r, E) = ψ(m)(r)−φ(r, E)

∫ r0 φ(x, E)ψ(m)(x)dx

1+∫ r

0

(φ(x, E)

)2dx

, m= 1,2, . . . . (259)

It is easy to show that

limr→0

˜ψ(m)3,2 (r) = lim

r→0ψ(m)(r) . (260)

Thus the normalization constants of the eigenfunctions are left unaltered. Eq. (218) canbe used to show that the singularity of the potential at the origin is unaltered after the twotransformations. It can also be shown that the phase shift relation for positive energies is

˜δ 3,2(l ,k) = δ (l ,k)+2 tan−1(

γk

). (261)

The expressions for the new potential and the new eigenfunctions [28] derived above arein agreement with the results derived using the Gelfand-Levitan method [26].

The family of potentials˜V3,2(r, E,α) in Eq. (253) lead to identical phase shifts and

the normalization constants for the excited states of the potentials˜V3,2(r, E,α) forvarious values ofα in the range0 < α < π/2, are the same. The parameterα affects

the normalization constant of the ground state. The family of potentials˜V3,2(r, E,α)have identical spectra, identical phase shifts and identical normalization constants forall states except the ground state and is, therefore, a phase-equivalent family. Thus atranformation of the typeT3 followed by a transformation of the typeT2 enables thegeneration of a family of potentials which have a new ground state but without alteringthe spectral density and without altering the singularity of the potential at the origin.

(5) We next considerT4 followed by T2. This sequence is very similar to case (4)except that in the first step involving the transformationT4 the Jost solutionη(r, E) isused instead of the regular solutionφ(r, E) in the potentialV. After theT4 transformationthe new potential is

V4(r, E) = V(r)− d2

dr2 lnη(r, E) . (262)

We now apply aT2 transformation at the energyE to introduce a new bound state belowthe ground state ofV. By identifying the general solution inV4 at energyE in the form

ψ4(r, E,β ) =cosβ +sinβ

∫ ∞r

(η(x, E)

)2dx

η(r, E), (263)

the new potential after the two transformations may be given in the form

˜V4,2(r, E,β ) = V(r)− d2

dr2 ln

(cosβ +sinβ

∫ ∞

r

(η(x, E)

)2dx

), (264)

which has a ground state eigenvalue atE while all the other eigenvalues are identical tothose ofV. The normalized ground state of˜V4,2 is given by

˜ψ(0)4,2(r, E,β ) =

(sinβ cosβ )1/2η(r, E)

cosβ +sinβ∫ ∞

r

(η(x, E)

)2dx

. (265)

Eq. (265) shows that the parameterβ determines the normalization constant of theground state. Forβ = π/4 the resultant potential after the two transformations is

˜V4,2(r, E) = V(r)− d2

dr2 ln

(1+

∫ ∞

r

(η(x, E)

)2dx

), (266)

which has the ground state eigenfunction

˜ψ(0)4,2(r, E) =

η(r, E)

1+∫ ∞

r

(η(x, E)

)2dx

. (267)

The excited state eigenfunctions are given by

˜ψ(m)4,2 (r) =−

(E(m)−E(0)

)−1A−2 A−4 ψ(m)(r) , m= 1,2, . . . , (268)

where

A−2 =1√2

[− d

dr+

ddr

ln

(1+

∫ ∞r

(η(x, E)

)2dx

η(r, E)

)],

A−4 =1√2

[− d

dr+

ddr

lnη(r, E)]

. (269)

Using the same methods as in the previous cases the excited state eigenfunctions can beshown to be given by

˜ψ(m)4,2 (r, E) = ψ(m)(r)−η(r, E)

∫ ∞r η(x, E)ψ(m)(x)dx

1+∫ ∞

r

(η(x, E)

)2dx

, m= 1,2, . . . . (270)

It can be shown that the normalization constants of the eigenfunctions are left unaltered.It can also be shown using Eq. (218) that the phase shifts are increased by a constantamountπ for all positive energies which is equivalent to saying that the phase shifts areunaltered within moduloπ. However

limr→0

˜V4,2(r) =(l −2)(l −1)

2r2 , limr→∞

˜V4,2(r) =l(l +1)

2r2 , (271)

which shows thatV4,2 has a component which has a short range attractiver−2 singularityin addition to the usual centrifugal potential.

Thus we have shown that the application of a transformation of the typeT4 followedby a transformation of the typeT2 enables the construction of singular potentials whichhave the same phase shifts as the original potential but which have an additional boundstate at a chosen energy. The family of potentials˜V4,2(r, E,β ), for various values ofβ inthe range0< β < π/2, have identical phase shifts and identical normalization constantsfor the excited states but have different normalization constants for the ground state andbelong to a family of phase-equivalent potentials.

(6) We now consider two successiveT3 transformations performed at the same energyE below the ground state ofV. After the firstT3 transformation using the regular solutionφ in the potentialV the new potentialV3 has the form given by Eq. (249). The secondT3transformation is performed using the choiceα = π/2 in Eq. (252) so thatψ3(r) = χ(r)which is defined by Eq. (251). The resulting potential after the two transformations isthen given by

˜V3,3(r, E) = V(r)− d2

dr2 ln

[∫ r

0

(φ(x, E)

)2dx

], (272)

which leads to a spectrum of eigenvalues which is identical with that forV. The trans-formed eigenstates may be shown to be given by

˜ψ(m)3,3 (r) = ψ(m)(r)−φ(r, E)

∫ r0 φ(x, E)ψ(m)(x)dx∫ r

0

(φ(x, E)

)2dx

, m= 0,1, . . . . (273)

It can be established that the normalization constants of the eigenfunctions and thespectral density are altered. Using Eq. (218) the phase shifts for positive energies canbe shown to be related by

˜δ 3,3(l ,k) = δ (l ,k)+2tan−1(

γk

)−π . (274)

It can also be shown that

limr→0

˜V3,3(r, E) =(l +2)(l +3)

2r2 , limr→∞

˜V3,3(r, E) =l(l +1)

2r2 , (275)

which shows thatV is a singular potential. Thus two successiveT3 transformations maybe used to produce a new singular potential which has the same spectrum asV but hasdifferent phase shifts and different normalization constants.

(7) We next considerT3 followed byT4. The first step is the same as in case (6). In thesecond step theT4 transformation is performed using the choiceα = 0 in Eq. (252) sothatψ3(r) = ξ (r) defined by Eq. (250). With this choice it is clear that the potential afterthe two transformations is identical to the original potentialV(r) sinceφ(r)ξ (r) = 1.ThusT4T3 restores the original Hamiltonian and all the eigenstates and phase shifts areunaltered. ThusT3 followed by T4 performed at the same energy is equivalent to theidentity transformation.

(8) The last combination we consider is two successiveT4 transformations at the sameenergyE below the ground state ofV. After the firstT4 transformation using the Jostsolutionη in the potentialV the new potentialV4 has the form given by Eq. (262). ThesecondT4 transformation is performed by using the choiceβ = π/2 in Eq. (263) so thatthe new potential after the two transformations is given by

˜V4,4(r, E) = V(r)− d2

dr2 ln

[∫ ∞

r

(η(x, E)

)2dx

], (276)

which has the same spectrum as that forV(r). The transformed eigenfunctions may beshown to be given by

˜ψ(m)4,4 (r) = ψ(m)(r)−η(r, E)

∫ ∞r η(x, E)ψ(m)(x)∫ ∞

r

(η(x, E)

)2dx

, m= 0,1, . . . . (277)

It can be established that the spectral density is altered and that the normalizationconstants of the eigenfunctions are altered. Using Eq. (218) it can be seen that the phaseshifts for positive energies are related by

˜δ 4,4(l ,k) = δ (l ,k)−2tan−1(

γk

)+π . (278)

It can also be shown that

limr→0

˜V4,4(r, E) =(l −2)(l −1)

2r2 , limr→∞

˜V4,4(r, E) =l(l +1)

2r2 , (279)

which shows thatV4,4 is a singular potential. Thus two successiveT4 transformationsmay be used to produce a new singular potential which has the same spectrum asV buthas different phase shifts and different normalization constants.

The new eigenfunctions after two transformations given by Eqs. (239), (244), (247),(259), (270), (273) and (277) can all be expressed as the ratio of 2 determinants. The newpotential can be expressed as a determinant of a matrix M. These identifications enablethe extension of the procedure outlined in this lecture to the elimination or addition of anarbitrary number of states or the generation of isospectral Hamiltonians after a numberof transformations at different energies. We now indicate the structure of the theory forthe seven nontrivial cases considered earlier.

For case (1) the matrix M has just a single element when a pair of transformations areused at a single energy and is given by

M11 = 1+λ∫ r

0

(ψ(0)(x)

)2dx . (280)

In terms of a second matrixM given by

˜M =(

M11∫ r

0 ψ(0)(x)ψ(m)(x)dxψ(0)(r) ψ(m)(r)

), (281)

the eigenfunction relation given in Eq. (239) may be expressed in the form

˜ψ(m)1,2 =

det ˜MdetM

. (282)

For case (2) the matrixM has element

M11 =∫ r

0

(ψ(0)(x)

)2dx , (283)

and ˜M is given by

˜M =(

M11∫ r

0 ψ(0)(x)ψ(m)(x)dxψ(0)(r) ψ(m)(r)

). (284)

The eigenfunction relation in Eq. (244) can be written in terms of these matrices as

˜ψ(m)1,3 =

det ˜MdetM

. (285)

For case (3) the matrixM has element

M11 =∫ ∞

r

(ψ(0)(x)

)2dx , (286)

and ˜M is given by

˜M =(

M11∫ ∞

r ψ(0)(x)ψ(m)(x)dxψ(0)(r) ψ(m)(r)

). (287)

The eigenfunction relation in Eq. (247) may then be written in the form

˜ψ(m)1,4 =

det ˜MdetM

. (288)

For case (4)M is given by

M11 = 1+∫ r

0

(φ(x, E)

)2dx , (289)

and the matrix ˜M has the form

˜M =(

M11∫ r

0 φ(x, E)ψ(m)(x)dxφ(r, E) ψ(m)(r)

). (290)

The eigenfunction relation in Eq. (259) may be given in the form

˜ψ(m)3,2 =

det ˜MdetM

. (291)

For case (5)

M11 = 1+∫ ∞

r

(η(x, E)

)2dx , (292)

and˜M =

(M11

∫ ∞r η(x, E)ψ(m)(x)dx

η(r, E) ψ(m)(r)

). (293)

The eigenfunction relation in Eq. (270) may be given in the form

˜ψ(m)4,2 =

det ˜MdetM

. (294)

For case (6)

M11 =∫ r

0

(φ(x, E)

)2dx , (295)

and˜M =

(M11

∫ r0 φ(x, E)ψ(m)(x)dx

φ(r, E) ψ(m)(r)

). (296)

The eigenfunction relation in Eq. (273) may be written in the form

˜ψ(m)3,3 =

det ˜MdetM

. (297)

Similarly for case (8)

M11 =∫ ∞

r

(η(x, E)

)2dx , (298)

and˜M =

(M11

∫ ∞r η(x, E)ψ(m)(x)dx

η(r, E) ψ(m)(r)

). (299)

The eigenfunction relation in Eq. (277) may be given in the form

˜ψ(m)4,4 =

det ˜MdetM

. (300)

5.3. Examples

The theory outlined in the last section may be illustrated with some exactly solvableexamples. To illustrate the first three cases we consider the potential

V(r) =−3sech2r , (301)

which supports a single bound state atE =−1/2 with eigenfunction

ψ(0)(r) =√

3Z√

1−Z2 , Z = tanhr . (302)

It is easy to show that∫ ∞

r

(ψ(0)(x)

)2dx = 1− tanh3 r , (303)

∫ r

0(ψ(0)(x))2dx = tanh3 r . (304)

By using the procedure for case (1) the family of phase-equivalent potentials in Eq. (233)can be given as

˜V1,2 =−3sech2r

(1+λ tanh3 r

)2

(1+2λ tanhrsech2r−λ 2 tanh4 r

). (305)

For values of∞ < λ <−1, the family of potentialsV1,2 have a single bound state at thesame energy but differ in the normalization constant of the ground state eigenfunction.λ = 0 corresponds to the original potentialV.

The method used for case (2) leads to the potential in Eq. (241) which in this examplebecomes

˜V1,3(r) =3

sinh2 r, (306)

which is repulsive singular potential which supports no bound states and is phase-equivalent toV(r).

The method used for case (3) leads to the potential in Eq. (246) which in this examplebecomes

˜V1,4(r) =−3

(sech2r

1+ tanhr + tanh2 r

)2

, (307)

which is an attractive non-singular potential which supports no bound states but has thephase shifts altered as specifed by Eq. (248).

We next examineV = 0 and the transformation energyE = −γ2/2. The regularsolution is

φ = sinhγr , limr→0

φ = γr , limr→∞

φ ∼ exp(+γr) , (308)

and the Jost solution is

η = exp(−γr) , limr→0

η = 1 , limr→∞

η = exp(−γr) . (309)

The potential arising from aT3 transformation is

V3 =γ2

sinh2 γr, (310)

which is a repulsive potential. The integral relation∫ r

0sinh2 γrdx≡ F =

sinh2γr−2γr4γ

, (311)

may be used to find the potential in Eq. (253) in the form

˜V3,2 =sinh4 γr

(cotα +F)2 −γ sinh2γrcotα +F

, (312)

which is a non-singular family of phase-equivalent potentials for0 < α < π/2 withnormalized ground state eigenfunctions given by

˜ψ3,2 =√

sinα cosαsinhγr

cosα +F sinα. (313)

The potential after theT4 transformation isV4 = 0. Using the integral relation∫ ∞

r(η(x))2dx=

exp(−2γr)2γ

, (314)

the potential in Eq. (264) in this example becomes

˜V4,2 =−γ2sech2(γr + ε) , 2ε = ln(2γ cotβ ) , (315)

which is also a family of phase-equivalent potentials for0< β < π/2, with ground stateeigenfunctions given by

˜ψ4,2∼exp(−γr)

cosβ +sinβ exp(−2γr). (316)

The potential after twoT3 transformations given by Eq. (272) in this example is

˜V3,3 =sinh4 γr

F2 − γF

sinh2γr , (317)

whereF is given by Eq. (311).The potential after twoT4 transformation given by Eq. (276) in this example gives

˜V4,4 = 0. This completes the list of transformed potentials after twoT transformations.

5.4. Summary

We have studied the properties of potentials and eigenfunctions arising from twosuccessive supersymmetric transformations performed at the same energy. We haveillustrated the procedure using an exactly solvable example. The method developed inthis lecture may be extended to eliminate or add an arbitrary number of states or alter thephase shifts without altering the spectrum. The generalized procedure for accomplishingthe above mentioned tasks will be the subject of the next lecture.

6. SEQUENCE OF SUSY TRANSFORMATIONS, REGULAR ANDSINGULAR POTENTIALS AND APPLICATIONS

The procedure discussed in the previous lectures for the modification of the spectrumof a potential corresponding to a radial Schrödinger equation by a two-step proceduremay be generalized to discuss the removal or addition of an arbitrary number of boundstates to a given spectrum or to generate the phase equivalent families correspondingto given potentials. The singular potentials arising from the two-step procedure haveproved useful in understanding the relation between deep and shallow potentials both ofwhich have been used with some success in describing a variety of scattering problemsin Nuclear Physics. Some examples will be discussed.

6.1. Elimination of m bound states and non-singular potentials

In §5 it was shown that a tranformation of the typeT1 followed by a transformationof the typeT4 may be used to remove the ground state of a potentialV(r) for adefinite partial wave without altering the spectral density and the normalization constantsof all the excited state eigenfunctions ofV(r). The phase shifts for positive energieswill, however, be altered in a definite manner. The resulting potential and the neweigenfunctions were given in terms of the matricesM and ˜M defined by

M11 =∫ ∞

r

(ψ(0)(x)

)2dx , (318)

and

˜M(n)

=

( ∫ ∞r

(ψ(0)(x)

)2dx

∫ ∞r ψ(0)(x)ψ(n)(x)dx

ψ(0)(r) ψ(n)(r)

), n = 1,2, . . . , (319)

in the form

˜V1,4(r) = V(r)− d2

dr2 ln(detM) (320)

˜ψ(n)1,4(r) =

det ˜M(n)

detM, n = 1,2, . . . . (321)

The phase shifts for positive energies were related by

˜δ 1,4(l ,k) = δ (l ,k)−2tan−1

(γ(0)

k

). (322)

It was also shown that the two transformationsT4T1 taken together do not introduce anyadditional singularities in the new potential.

The procedure used for eliminating a single bound state may be generalized to thecase of elimination of an arbitrary numberm of bound states as follows. LetV0 be apotential with bound states given by

V0 :

[E(1) E(2) . . . E(n)

ψ(1)0 ψ(2)

0 . . . ψ(n)0

], (323)

determined from the solutions to the radial Schrödinger equation[−1

2d2

dr2 +l(l +1)

2r2 +V0(r)−E( j)]

ψ( j)0 (r) = 0 , j = 1,2, . . . ,n . (324)

The eigenfunction set in Eq. (323) may be taken to be an orthonormal set. The incom-plete overlap integrals may be defined by

Fjk =∫ ∞

rψ( j)

0 (x)ψ(k)0 (x)dx , j,k = 1,2, . . . ,n , (325)

G jk =∫ r

0ψ( j)

0 (x)ψ(k)0 (x)dx , j,k = 1,2, . . . ,n . (326)

The two matricesF andG are related byF +G = I whereI is the unit matrix of dimen-sion [n,n]. By using the orthonormal property of the eigenfunctions and the Wronskianrelation between two solutions of the Schrödinger equation in the same potential at twodifferent energies,Fjk andG jk can also be written as

Fjk =−G jk =1

2(E(k)−E( j)

)(

ψ( j)0 (r)

ddr

ψ(k)0 (r)−ψ(k)

0 (r)ddr

ψ( j)0 (r)

), j 6= k .

(327)Let Vm be the potential generated fromV0 by the application of a sequence consisting

of m pairs ofT1 andT4 transformations for eliminating the lowestm bound states ofV0so that the bound states ofVm are given by

Vm :

[E(m+1) E(m+2) . . . E(n)

ψ(m+1)m ψ(m+2)

m . . . ψ(n)m

]. (328)

Then it may be shown by a generalization of the method for eliminating a single boundstate that in terms of a matrixM with elements given by

M jk = Fjk , j,k = 1,2, . . . ,m , (329)

the potentialVm with n−mbound states may be given as

Vm = V0− d2

dr2 ln(detM) . (330)

The eigenfunctions ofVm are related to those ofV0 by

ψ(i)m =

det ˜M(i)

detM, i = m+1,m+2, . . . ,n , (331)

where the elements ofM(i)

are

˜M(i)jk = M jk , j,k = 1,2, . . . ,m ,

˜M(i)j,m+1 =

∫ ∞

rψ( j)

0 (x)ψ(i)0 (x)dx= Fji , j = 1,2, . . . ,m ,

˜M(i)m+1,k = ψ(k)

0 (r) , k = 1,2, . . . ,m ,

˜M(i)m+1,m+1 = ψ(i)

0 (r) , i = m+1,m+2, . . . ,n . (332)

Eqs. (329)-(332) provide a recipe for constructing a new potential by removingmbound states and which does not have any additional singularities present which are notalready present inV0. This recipe is the same as the recipe based on the Gelfand-Levitanequations. The phase shifts in the new potentialVm for positive energies are related tothe phase shifts inV0 by

˜δ m(l ,k) = δ (l ,k)−2m

∑j=1

tan−1

(γ( j)

k

), (333)

whereγ( j) =√−2E( j). In the limit k→ 0 the phase shift decreases bymπ in agreement

with the general result for a potential withm fewer bound states. In the limitk→ ∞ thephase shifts in the two potentials are identical.

The expression for the new eigenfunctions in Eq. (331) may also be given in otherforms by expanding the determinant in the numerator and using the definition of theelements of the inverse of a matrix as the ratio of two determinants. This leads to theexpression

ψ(i)m = ψ(i)

0 −m

∑k=1

m

∑j=1

ψ(k)0

(M−1)

k j Fji , i = m+1,m+2, . . . ,n . (334)

Since the indexj in the above equation can only take values in the range1≤ j ≤mandthe indexi takes valuesm< i ≤ n the matrix elementFji may be replaced by−G ji sothat Eq. (334) may also be given in the form

ψ(i)m = ψ(i)

0 +m

∑k=1

m

∑j=1

ψ(k)0

(M−1)

k j G ji , i = m+1,m+2, . . . ,n . (335)

It may be shown that in this form the above equation may be extended to allow the indexi to take all values in the range[1,n] to define a set of solutions inVm at the energies atwhich the bound states have been removed. It can be shown that these non-normalizablesolutions inVm at the energies corresponding to the removed bound states satisfy

ψ( j)m =

m

∑k=1

ψ(k)0

(M−1)

k j . (336)

In terms of the solutions to the linear equations

m

∑j=1

Mk jψ( j)m = ψ( j)

0 , k = 1,2, . . . ,m . (337)

the normalizable bound state eigenfunctions ofVm may be given as

ψ(i)m = ψ(i)

0 −m

∑k=1

ψ(k)m Fki , i = m+1,m+2, . . . ,n . (338)

Thus by constructing the matrixM and solving the linear equations (337) to findψ(k)0

once it is possible to find all the eigen functions ofVm using Eq. (338). It may beshown that the expression for the new potential in Eq. (330) can be further simplified byexpanding the logarithmic derivative to get

ddr

ln(detM) =m

∑k=1

ψ(k)0 (r)ψ(k)

m (r) , (339)

so that the modified potential may be expressed in the form

Vm(r) = V0(r)− ddr

(m

∑k=1

ψ(k)0 (r) ψ(k)

m (r)

). (340)

In this form the potential arising from the removal ofm bound states is expressed as asymmetric function of the solutions in the old and new potentials at the energies at whichthe bound states have been removed.

6.2. Elimination of m bound states and singular potentials

In §5 it was shown that a transformation of the typeT1 followed by a transformationof the typeT3 may be used to remove the ground state of a potentialV(r) withoutaltering the phase shifts for positive energies. However, the normalization constants ofthe eigenstates and the spectral density are altered at the end of the two transformations.The resulting potential and the new eigenfunctions were given in terms of the matricesN and ˜N defined by

N11 =∫ r

0

(ψ(0)(x)

)2dx , (341)

and

˜N(n)

=

( ∫ r0

(ψ(0)(x)

)2dx

∫ r0 ψ(0)(x)ψ(n)(x)dx

ψ(0)(r) ψ(n)(r)

), n = 1,2, . . . , (342)

in the form

˜V1,3(r) = V(r)− d2

dr2 ln(detN) , (343)

˜ψ(n)1,3(r) =

det ˜N(n)

detN, n = 1,2, . . . . (344)

It was also shown thatV1,3 is a singular potential with a short range repulsiver−2

character which asr → 0 is equivalent tol → (l + 2) in the centrifugal part of thepotential. It was also shown that the phase shift decreases byπ for all energies andtherefore the phase shifts in the two potentials are the same within moduloπ.

The procedure used for eliminating a single bound state may be generalized to thecase of elimination of an arbitrary numberm of bound states. LetVm be the potentialgenerated fromV0 by the application of a sequence consisting ofm pairs ofT1 andT3transformations for eliminating the lowestm bound states ofV0 so that the bound statesof Vm are given by

Vm :

[E(m+1) E(m+2) . . . E(n)

ψ(m+1)m ψ(m+2)

m . . . ψ(n)m

]. (345)

It may be shown by a generalization of the method for eliminating a single bound statethat in terms of a matrixN with elements given by

Njk =∫ r

0ψ( j)

0 (x)ψ(k)0 (x)dx= G jk , j,k = 1,2, . . . ,m , (346)

the potentialVm with n−m bound states may be given as

Vm = V0− d2

dr2 ln(detN) . (347)

It is clear that the elements of the matrixN and the matrixM considered in the previoussection are related byM + N = I whereI is the unit matrix of dimension(m,m). Thismeans that the elements of the two matrices are related by

Njk =−M jk , j 6= k , Nj j = 1−M j j . (348)

The eigenfunctions ofVm are related to the eigenfunctions ofV0 by

ψ(i)m =

det ˜N(i)

detN, i = m+1,m+2, . . . ,n , (349)

where the elements ofN(i)

are

˜N(i)jk = Njk , j,k = 1,2, . . . ,m ,

˜N(i)j,m+1 =

∫ r

0ψ( j)

0 (x)ψ(i)0 (x)dx= G ji =−Fji , j = 1,2, . . . ,m ,

˜N(i)m+1,k = ψ(k)

0 (r) , k = 1,2, . . . ,m ,

˜N(i)m+1,m+1 = ψ(i)

0 (r) , i = m+1,m+2. . . ,n . (350)

Eqs. (346)-(350) provide a recipe for constructing a new potential by removingmboundstates without altering the phase shifts but altering the spectral density. In fact the phaseshifts decrease bymπ for all positive energies but the two phase shifts are the samewithin modulo π. The resulting potential is singular and has a repulsive character asr → 0. It can be shown that

limr→0

Vm =(l +2m)(l +2m+1)

2r2 , limr→∞

Vm =l(l +1)

2r2 , (351)

which shows that the repulsive singularity at the origin rises rapidly as the number ofbound states removed increases.

The expression for the new eigenfunctions in Eq. (349) may also be given in otherforms by expanding the determinant in the numerator and using the definition of theelements of the inverse of a matrix as the ratio of two determinants. This leads to theexpression

ψ(i)m = ψ(i)

0 +m

∑k=1

m

∑j=1

ψ(k)0

(N−1)

k j Fji , i = m+1,m+2, . . . ,n . (352)

It may be shown that the non-normalizable solutions inVm at the energies correspondingto the removed bound states may also be defined by Eq. (352) by allowing the indexito take values in the range(1,m). Then by usingFjk = M jk and the matrix relation inEq. (344) the above equation may be further simplified to give

ψ( j)m =

m

∑k=1

ψ(k)0

(N−1)

k j , j = 1,2, . . . ,m . (353)

Thus in terms of the solutions to the linear equations

m

∑k=1

Njkψ(k)m = ψ( j)

0 , j = 1,2, . . . ,m , (354)

the bound state eigenfunctions inVm are

ψ(i)m = ψ(i)

0 −m

∑k=1

ψ(k)m Gki , i = m+1,m+2, . . . ,n . (355)

Thus by constructing the matrixN and solving the linear equations (354) to findψ(k)m

once it is possible to find all the eigen functions ofVm using Eq. (355). It may beshown that the expression for the new potential in Eq. (347) can be further simplified byexpanding the logarithmic derivative to get

ddr

ln(detN) =m

∑k=1

ψ(k)0 (r)ψ(k)

m (r) , (356)

so that the modified potential may be expressed in the form

Vm(r) = V0(r)− ddr

(m

∑k=1

ψ(k)0 (r)ψ(k)

m (r)

). (357)

In this form the potential arising from the removal ofm bound states is expressed as asymmetric function of the solutions in the old and new potentials at the energies at whichthe bound states have been removed.

So far we have discussed in detail the procedure for removingm bound states toproduce singular or non-singular potentials. It is possible to give similar generalizationsof the other cases discussed in lecture IV to provide recipes for addingmbound states toproduce singular and non-singular potentials or find extended families of potentials withthe same spectrum but differing in the normalization constants of the eigenfunctionsand/or differing in the phase shifts. We have concentrated on a detailed discussion of thecase of the removal bound states because this has found some applications in NuclearPhysics. We discuss the application of these ideas to certain problems in Nuclear Physicsin the next section.

6.3. Deep and shallow potentials in nuclear physics

In Nuclear Physics both deep and shallow potentials have been used to describenucleus-nucleus interactions both of which fit the set of known experimental data suchas energy levels equally well. However, their predictions can differ for other energydomains where the experiments have not been performed yet or where the propertiesstudied involve the explicit use of eigenfunctions to calculate matrix element suchas the calculation of electromagnetic transition probabilities or radiative-capture crosssections. The question of the relation between deep and shallow potentials and theirappropriateness for the study of nuclear interactions has been a controversial issue for along time.

Microscopic models making use of fully antisymmetric scattering eigenfunctionswhich provide phase shifts that satisfy a modified Levinson’s theorem [49, 50, 51] havebeen developed [52] which lead to a description of internuclear interactions in termsof deep local potentials. These deep potentials have a number of unphysical boundstates which simulate the forbidden states of a microscopic approach to the study ofnucleus-nucleus interaction and are needed to provide phase shifts which agree with thegeneralized Levinson theorem in the high energy limit. Deep potentials which accuratelyfit the data forα +α (Bucket al.[52]), α+16O (Michelet al.[53]) andα+40Ca (Michel

and Vanderpoorten [54]) have been constructed. The real part of these potentials eitherdo not depend or weekly depend on energy and angular momentum.

In contrast shallow potentials which have no unphysical bound states in their spectrabut which have strong angular momentum dependence and fit the same experimentaldata equally well have also been constructed for some systems such asα + α (Aliand Bodmer [55]). The bound states of the shallow potentials can be interpreted asgood approximations to the actual physical states of the fused nucleus. Michel andReidmeister [56] showed that it is possible to construct shallow potentials which aresingular whenr → 0 but are able to produce phase shifts at high energies which arein agreement with the microscopic phase shifts and derived phenomenological singularpotentials which are phase equivalent to the real part of theα +16O deep potential.

The relation between inverse scattering theory and the algebra of supersymmetrystudied in these lectures sheds light on the relation between deep and shallow potentialsboth of which explain the same data well. The method discussed in §6.3 to removea number of bound states from a deep potentialV0 and generate a singular potentialVm with m fewer bound states but which is phase equivalent to the deep potentialhas been used for a number of nuclear systems by Baye [57]. We now consider theα + α scattering discussed by Baye. The starting potentialV0 for this system is thetwo parameter potential of Buck, Friedrich and Wheatley [52]. The s-wave potentialV0 has three bound states one of which is at .092 MeV and corresponds to a physicalstate, (viz) the ground state ofBe8, and two other bound states at -72.8 MeV and -25.9MeV which are unphysical. The two-step SUSY procedure for removing bound statesdiscussed in §6.3 is used to first remove the bound state at -72.8 MeV to produce thesingular potentialV1 which has two bound states. The potential arising from the removalof both the unphysical bound states ofV0 is the potentialV2 which is singular but hasonly one bound state which is the physical bound state.V0, V1 andV2 have identicalphase shifts for all energies and are phase equivalent. Fig. 8 shows the results of sucha calculation similar to the one performed by Baye [57]. TheV2 so constructed is theshallow but singular potential which is the phase equivalent partner fors-waves to thedeep potential used by Bucket al.. The deep potential has one unphysical bound statefor l = 2 and no unphysical bound states forl = 4. The SUSY procedure was used byBaye to remove the unphysical bound states and the resulting potentials forl = 0,2 and4 partial waves denoted byV(0)

2 , V(2)1 andV(4)

0 , respectively, were compared with theshallowα +α potentials used by Ali and Bodmer [55] and good agreement was found.

Thus the calculation by Baye [57] has demonstrated that the singular, shallow,l -dependent Ali-Bodmer potentials forα +α scattering is an approximate supersymmetricpartner to the deep,l -independent potential of Buck, Friedrich and Wheatley.

Sparenberg and Baye [58] have analysed inverse scattering with singular potentials indetail using the supersymmetric approach. They have shown that by using potentialswith a r−2 singularity at the origin, the inverse scattering problem at fixed orbitalangular momentuml can be decomposed into two parts. In the first step the phaseshift information is used to construct a singular potential without a bound state. Whenthe phase shift at higher energies is smaller than the phase shift at zero energy, theeffective potential has a repulsive core of the formν(ν + 1)/(2r2) whereν is largerthanl . In the second step the bound states are added without modifying the phase shift.

FIGURE 8. Singularα +α potentials forl = 0 generated from the deep potentialV0 of Buck, Friedrichand Wheatley [52] by the successive elimination of the two unphysical bound states ofV0. V2 is phaseequivalent toV0 andV1. The removed bound states are represented by horizontal bars.

Sparenberg and Baye have used this approach to invert the experimental phase shifts ofneutron-proton elastic scattering in the1S0 state to obtain singular potential withν = 1.Andrianov, Borisov and Ioffe [59] have investigated the possibility of a supersymmetricdescription of two half-integer spin particles by studying the supersymmetric aspect ofNN andNN scattering in the one-pion exchange approximation. Andrianov, Cannata,Ioffe and Nishanidze [60] have studied matrix Hamiltonians using a SUSY approachto hidden symmetries and have analysed coupled channel problems using this approach.Sparenberg and Baye [61] have used SUSYQM to find coupled-channel potentials whichare phase equivalent and have removed the non-physical state of the deep3S1-3D1neutron-proton Moscow potential and transformed it into a shallow potential with arepulsive core. There are many other examples of such applications of singular potentialsin Nuclear Physics.

7. CONCLUSIONS

In this course of lectures we have shown that starting from the simple idea of factorisingsecond order differential equations of a single variable it is possible to find a relateddifferential equation whose solutions are closely related to those of the original equation.

The relationship between the spectra and eigenfunctions of the two differential operatorsare similar to those between the components of a supersymmetric doublet. The flexibilityavailable in the choice of factorization enables the manipulation of the spectrum of theoperators in desired fashion. We have shown that using the factorization method it ispossible to add or remove bound states or maintain the same spectrum but alter thenormalization constants of the eigenfunctions.

We have used these ideas to study the radial Schrödinger equation in detail. If the po-tential of interaction is known then it is possible to find the eigenvalue spectrum, eigen-functions and the phase shifts for positive energies by solving the radial Schrödingerequation for a given partial wave. But if the potential of interaction is not known ac-curately from first principles then one must use the spectral data like phase shifts andenergy levels to work backwards to find the potential. This is the subject of study ofinverse scattering theory. To determine the potential uniquely all the energy levels, thenormalization constants associated with the eigenfunctions of all the energy levels andthe phase shifts at all positive energies from0 to ∞ must be known. Such a complete listof data seldom exists and often phenomenological potentials with widely differing prop-erties in terms of their behaviour as a function ofr are used. In these lectures we haveshown that the concept of supersymmetric pairing enables the construction of inversescattering theory from some basic building blocks. We have shown that it is possibleto construct families of potentials with related phase shifts and other spectral proper-ties. Such constructions have enabled the establishment of links between potentials withstartlingly dissimilar structure which nevertheless have closely related spectral proper-ties. Even though the same properties could be found from the usual methods of inversescattering theory based on Gelfand-Levitan and Marchenko equations the method basedon the methods of SUSYQM has a particular simplicity and economy of expression andopens up the possibility of similar constructions for other inverse problems arising inother contexts. The method based on the supersymmetric construction has now beenextended to the study of coupled differential equations which has proved useful in thestudy of a variety of coupled-channel problems in Nuclear Physics.

We have shown that the concept of supersymmetric pairing may be used to elucidatethe spectrum of the Dirac equation for an attractive Coulomb potential. We have alsoshown that the Backlund transformation which provides a link between theN andN+1soliton solutions of the KdV hierarchy of non-linear differential equations is closelyrelated to a supersymmetric transformation which links the potentialVN with N boundstates to another potentialVN+1 which has an additional bound state below the groundstate ofVN.

The construction of singular potentials using suitable pairs of supersymmetric trans-formations and their success in elucidating the relationship between diffferent modelpotentials in Nuclear Physics is an outstanding example of the usefulness of SUSYQMin a real physical context.

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