Supersymmetric Transformations and the Inverse Problem in Quantum

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Universit´e Libre de Bruxelles

Facult´e des Sciences appliqu´ees

Supersymmetric Transformations and the Inverse Problem in Quantum

Mechanics

Thèse présentée en vue de l’obtention du titre de Docteur en Sciences appliquées par

Jean-Marc Sparenberg

Janvier 1999

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Remerciements

Je tiens avant tout à remercier chaleureusement le Professeur Daniel Baye pour son encadrement de qualité et sa disponibilité tout au long de cette thèse. Tant ses qualités de chercheur que ses qualités d’enseignant sont un exemple permanent pour moi.

Ces années de recherche se sont déroulées au sein du Service de Physique Nucléaire Théorique et de Physique Mathématique de la Faculté des Sciences.

Je remercie le chef de ce service, le Professeur Christiane Leclercq-Willain, pour son accueil et pour les possibilités qu’elle m’a offertes d’assister à de nombreuses conférences et écoles, tant en Belgique qu’à étranger.

Mes collègues et amis, membres de ce même service, m’ont souvent été d’une aide précieuse : je pense en particulier à Pierre Descouvemont pour son assistance informatique, à Guy Reidemeister pour ses conseils bibliogra- phiques, et à Michel Hesse pour les calculs qu’il a réalisés, tant sur les noyaux

`a halo que sur les potentiels de Bargmann en voies coupl´ees. Je leur exprime ici toute ma gratitude.

Ma formation de chercheur a été enrichie à deux reprises par des colla- borations internationales. Avec Géza Lévai (Debrecen, Hongrie), nous avons

étudié la construction de potentiels analytiquement solubles par transforma- tions de supersymétrie. Le Professeur Bunryu Imanishi (Tokyo, Japon) m’a permis d’utiliser son programme de collisions en voies couplées. Je le remercie pour le temps important qu’il a consacré à me guider dans l’utilisation de ce programme, et à commenter (par voie électronique) les résultats obtenus.

Sur un plan plus personnel, je tiens également à remercier tous ceux qui m’ont soutenu au cours de ces quatre années, et plus particulièrement au cours de la rédaction. Je pense tout d’abord à Aline, dont les talents de plafonneuse, de peintre, d’électricienne, d’entrepreneuse, de décoratrice, et de cuisinière, m’ont presque fait oublier qu’une certaine maison était encore

`a r´enover. Plus fondamentalement, je la remercie pour sa tendresse et son

´ecoute patiente tout au long de mon travail.

Viennent ensuite tous ceux qui, d’une fa¸con ou d’une autre, ont mani- festé leur intérêt si pas pour mon travail, au moins pour mon mode de vie ! Parmi ceux-ci, mes parents et mes proches, mais aussi tous les amis qui ont un jour osé poser (ou reposer) une des questions fatidiques : “Que fais-tu encore exactement ?”, “Et ¸ca sert à quoi ?”, “La mécanique quan . . .quoi ?”,

“Et le titre de ta thèse (qu’on rigole) ?”. Je suis conscient que mes réponses n’ont pas toujours satisfait à leurs attentes, et je les remercie de m’avoir par ce biais incité à une plus grande humilité . . .

La première année de ce travail a été financée dans le cadre d’un programme

de Pôle d’Attraction Interuniversitaire. J’ai ensuite bénéficié d’un mandat

d’Aspirant du Fonds National de la Recherche Scientifique qui a ´et´e recon-

duit pour deux ans. Je remercie ces deux organismes pour leur financement.

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Chapter 1 Introduction

1.1 Supersymmetric quantum mechanics

Introduced in the early eighties [Wit81] as a simple example of the supersym- metric algebra used in field theory, supersymmetric quantum mechanics has rapidly become a subject of interest on its own [Suk85b]. Meanwhile, it has also been realized [Nie84, ABI84] that its formalism was actually equivalent to the so-called Darboux transformations, proposed in 1882 in Ref. [Dar82a], and to the Schr¨odinger factorization method proposed in 1940 in Ref. [Sch40].

In the following, we shall use indiscriminately the terms “supersymmetric”

or “Darboux”.

The principle of these three equivalent methods can be explained in simple words. Let us consider a Schr¨odinger stationary equation in one dimension, containing a given interaction potential. We assume that the solutions of this second-order differential equation (bound states, scattering states, and even non-physical solutions) are known, either numerically or analytically. Then, by applying an operator to this initial equation, we transform it into a new Schr¨odinger equation, with a new interaction potential. The supersymmetric formalism implies that all the solutions of this new equation are expressed by algebraic expressions in terms of the solutions of the initial equation.

Moreover, the new potential is expressed in terms of the initial potential and of one particular solution of the initial equation, called the factorization solution. Different factorization solutions provide different transformed po- tentials, but the algebraic structure of the method does not depend on the particular choice of factorization solution. In summary, it can be said that supersymmetric transformations allow the construction of new Schr¨odinger equations, starting from a given one, and that everything known about the initial equation is also known about the new ones.

In the present work, we consider radial Schr¨odinger equations. These appear in the quantum description of two interacting particles, either form- ing bound states at negative energy, or colliding at relatively low positive energies (non-relativistic collisions) [Tay72]. In such a description, an im- portant quantity is the Jost function, from which physical observables are

1

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calculated

¹

: bound states are related to its zeroes and collision cross sections are related to its phase. Through a supersymmetric transformation, this Jost function is modified in a simple but powerful way: roughly-speaking, it is multiplied (or divided) by a rational factor [Suk85a]. This leads to two types of applications:

• supersymmetric transformations can add, remove or modify bound states by acting on the zeroes of the Jost function;

• they can modify cross sections by acting on the phase of the Jost func- tion.

These two features make of supersymmetric or Darboux transformations an ideal tool to address the so-called inverse problem.

1.2 Application to the inverse problem

The purpose of the inverse problem [CS77] is to construct an interaction potential from scattering data (cross sections). Actually, in the presence of bound states, the potential cannot be uniquely determined from scattering data: information about the bound states is also needed. In other words, bound states create an ambiguity in the inverse problem. This suggests to treat both aspects of the inverse problem separately: first, to construct one potential from scattering data (which, in the traditional meaning of the term, constitutes the “inverse problem” itself); second, to construct all the other potentials sharing the same cross sections as this potential but with different bound-spectrum properties. The aim of this work is to show that such a decomposition of the inverse problem is in fact possible with supersymmetric transformations.

Historically, our group has first used the supersymmetric formalism to address the second part of this decomposition, namely the construction of so- called phase-equivalent potentials [Bay87b, Bay87a], i.e. of potentials which have identical scattering properties. An important remark has to be done about this term. In the literature, “phase equivalent” is sometimes used for potentials which are constrained to have the same bound spectrum or at least the same number of bound states. In this work, important results are obtained with phase-equivalent potentials which have different number of bound states. We thus use the term “phase equivalent” in the most gen- eral sense and reserve the term “isospectral” for potentials sharing the same spectrum.

Such phase-equivalent potentials can be constructed with the help of two successive supersymmetric transformations [Bay87b, Bay87a]. The possibil- ity of removing or adding bound states without modifying the scattering ma- trix is particularly interesting from the physical point of view, since it allows

1Better known than the Jost function is the scattering operator or collision matrix.

Actually, this collision matrix is deduced in a unique way from the Jost function, whereas we shall see that different Jost functions may correspond to the same collision matrix.

The Jost function thus contains more physical information than the collision matrix.

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1.3. ORGANIZATION OF THE WORK 3 a connection between very different families of phenomenological potentials.

Actually, in nuclear physics, the coexistence of deep and shallow phenomeno- logical potentials is a well-known ambiguity of the inverse problem. Shallow potentials have physical bound states only, while deep potentials have addi- tional forbidden states [Hor91], simulating the effect of the Pauli exclusion principle between the constituents of the colliding particles (nucleons in the case of a nucleus-nucleus collision). Both types of potentials have very dif- ferent form factors, and it is rather surprising at first sight that they provide the same or similar cross sections. However, such a phase equivalence can be explained with the aid of supersymmetric transformations, as shown for instance for the α+α collision in Ref. [Bay87b].

In this work, we generalize the supersymmetric formalism to various classes of potentials like complex (optical) potentials, coupled-channel po- tentials, and energy-dependent potentials. These generalizations are very useful for physical applications, since they are necessary as soon as collisions are considered for which the internal structure of the particles is modified during the scattering process (inelastic collisions and reactions). The study of phase-equivalent potentials will thus be extended to these cases.

The first part of the decomposition of the inverse problem proposed here, namely the construction of one interaction potential starting from scatter- ing data, is a long-standing problem in quantum mechanics [CS77]. It is generally addressed with the aid of a sophisticated mathematical formalism, leading to integral equations. Various methods have been proposed to ap- proximately solve these equations [CS77]. It was known from the beginning [Nie84, Suk85a] that supersymmetric or Darboux transformations also led to efficient approximate methods to solve them, but it has even been demon- strated recently [SL94] that most of the existing resolution methods of these integral equations can actually be formulated in terms of pairs of Darboux transformations.

In the present work, we take up this first part of the inverse problem with two innovative aspects. First, we use potentials which may be singular at the origin; this allows us to construct potentials without bound state, and hence without ambiguity. Second, we use single supersymmetric transformations (rather than pairs); this leads to new classes of potentials, which could not be found with existing methods.

To summarize, one sees that supersymmetric transformations provide a comprehensive tool to solve the inverse problem, since they allow both the construction of an interaction potential from scattering data, and the treat- ment of the ambiguities in this potential due to the presence of bound states.

1.3 Organization of the work

Chapter 2 contains a summary of the non-relativistic collision theory. The

accent is put on the coupled-channel aspects, as well as on potentials singu-

lar at the origin. Another important point discussed in this chapter is the

respective roles of the Jost function and of the scattering matrix.

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Chapter 3 introduces the supersymmetric transformations. An historical introduction first establishes the link between the three equivalent meth- ods mentioned above (Darboux transformations, Schr¨odinger factorization method, and supersymmetric quantum mechanics). Then the formalism is explicitly derived for a generalized Schr¨odinger equation, which allows a uni- fied treatment of both the fixed-angular-momentum and the fixed-energy cases [RSZ84], as well as the introduction of a linear energy or angular- momentum dependence in the potential [SL93]. Important formulae for fur- ther applications (scattering-matrix modification and pairs of supersymmet- ric transformations) are then derived.

For each example or application presented in the rest of the work, we first rewrite the important formulae (without proving them any more) in a simplified form for the considered particular case, and then only concentrate on physical results and interpretations. Our hope is that each remaining chapter (4 to 6) is self-contained and may be read independently, provided some basic formulae proved in Chap. 3 are accepted.

Chapter 4 is devoted to the treatment of bound states through supersym- metric transformations. The phase-equivalent pairs are first presented, with particular emphasis on the bound-state removal, which is sufficient to estab- lish links between deep and shallow potential families. Then it is shown on a example that supersymmetric transformations provide all the potentials sharing a given scattering matrix in the coupled-channel case, at least for a particular class of scattering matrices. The occurrence of degenerate and non-degenerate bound states is studied.

Chapter 5 aims at comparing the physical properties of deep and shal- low potentials, which are frequently encountered in nuclear physics. A short discussion of the physical interpretation of the forbidden states appearing in deep potentials is first made. Then a supersymmetric connection between po- tential families is established and the potentials are used in various models in order to compare their properties in other contexts than two-body collisions.

This is done for different physical systems corresponding to different types of potentials: the α+neutron system for real energy-independent potentials, with an application to a three-body calculation of the

⁶

He halo nucleus; the

16

O+

¹⁶

O system for complex potentials, with an application to a three-body calculation of the

¹⁷

O+

¹⁶

O collision; the α+

¹⁶

O system for energy-dependent potentials; and the neutron+proton system for matrix potentials.

Chapter 6 is devoted to supersymmetric inversion methods, both in the fixed-energy and fixed-angular-momentum cases. The interest of a single- transformation approach is demonstrated. Moreover, a new approach to the fixed-angular-momentum inverse problem, based on the use of potentials sin- gular at the origin, is proposed and applied to the nucleon+nucleon system.

Chapter 7 contains our conclusions and perspectives.

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Chapter 2 Non-Relativistic Scattering Theory

In this chapter, we summarize the non-relativistic time-independent scatter- ing theory and study the radial coupled-channel (matrix) Schr¨odinger equa- tion, on which supersymmetric transformations will be performed in the fol- lowing. For more detailed presentations, we refer to Refs. [Joa83, New82, Tay72]. We first specify our hypotheses (Sec. 2.1), particularly insisting on the possibility of potentials presenting an r

⁻²

repulsive singularity at the ori- gin; such potentials will be used in many applications in this work. Then we detail in Sec. 2.2 some important notions specific to the coupled-channel case (Wronskian of matrix functions, linear independence and self conjugation), and deduce some useful properties of the solutions of the Sch¨odinger matrix equation (Sec. 2.3). Finally, we introduce different types of solutions of the Schr¨odinger equation which will be used in the following; mathematical solu- tions are defined in Sec. 2.4 by their boundary behaviour, either at the origin or at infinity, and are used in Sec. 2.5 to construct physical solutions (bound and scattering states). We particularly insist on the relation between the Jost function and the scattering matrix, and on the fact that only the Jost function provides unambiguous information about the occurrence of bound states.

2.1 Hypotheses

2.1.1 Non-relativistic time-independent treatment

The energy of the colliding particles is low enough for the relativistic effects not to play any role. Consequently, the Schr¨odinger wave equation can be used, rather than the Klein-Gordon or Dirac equations. The only relativistic effect which can be taken into account in the formalism explained below is the spin of the particles; this is done by generalizing the traditional geometric Hilbert space to a tensor product of a spin space and a geometric space. The particles are then described by spinors, which can be considered as multi- channel wave functions (see Subsec. 2.1.3).

5

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We directly adopt the time-independent version of the Schr¨odinger equa- tion, which simplifies all calculations without loosing any physical content.

This equation reads

HΨ = EΨ, (2.1)

where E is the energy of the system, Ψ is a stationary wave function de- pending on the coordinates of all the particles, and H is the Hamiltonian operator, composed of the kinetic-energy operators of the particles and of their interaction potentials. It is certainly a remarkable feature of quantum theory that time seems to “disappear” from all the equations, being like an unnecessary quantity. This feature is even more striking in collision theory, where in a classical view time seems to be essential

¹

.

2.1.2 Coupled-channel approximation, optical poten- tial

We only consider collisions which can be satisfactorily described with a lit- tle number N of two-body channels. These channels are characterized by the mass partition between the two bodies, and by the internal structure (charge, energy, spin, and so on) of these two bodies. This implies that the coupled-channel or close-coupling approximation can be used (Sec. 19-c of Ref. [Tay72]), and that the general Schr¨odinger equation describing the sys- tem reduces to a system of N coupled equations with only one coordinate r describing the relative motion of the particles

²

,

h

−∇

²

+ V (r)

ⁱ

Ψ(K, r) = K

²

Ψ(K, r), (2.2) with the unit choice ¯ h = 1. In this equation (see Subsec. 17.1.1 of Ref.

[New82]), K is a diagonal matrix, the elements of which are the wave numbers of the channels, which can be deduced up to a sign from the energy and the thresholds by

K

ⁱ

= ±

^q

2µ

ⁱ

(E − ∆

ⁱ

) (i = 1, . . . , N ), (2.3) where µ

ⁱ

is the reduced mass of channel i

³

, and ∆

ⁱ

is the threshold energy of channel i with respect to channel 1. We agree upon channel 1 being the

1Historically, the time-independent (or stationary) formalism was introduced in quantum mechanics before the time-dependent one. This is because the bound-state description was first addressed, which uses negative-energy stationary states. In the late twenties, calculations with positive-energy stationary states were then found sufficient to describe scattering experiments, even if they raised some interpretation problems like infinitely-spread wave packets. These problems were solved in the late fifties, when the time-independent formalism was deduced from the time-dependent one. A pedagogic presentation of this deduction is given in Ref. [Tay72].

2We exclude the possibility of non-local potentials.

3This formalism is thus valid for channels with different reduced masses (rearrangement collisions). However, we shall see in Subsec. 3.2.3 that supersymmetric transformations seem to be only applicable to equations with equal reduced masses.

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2.1. HYPOTHESES 7 channel were the internal energy of the colliding particles is the lowest, and choose as zero energy the energy of the system in this channel when the particles are infinitely separated, i.e.

∆

¹

= 0 ≤ ∆

²

≤ . . . ≤ ∆

^N

. (2.4) Equation 2.3 can be put in matrix form

K

²

= 2µ(E − ∆), (2.5)

where µ and ∆ are diagonal matrices formed by the reduced masses and the thresholds respectively. This equation expresses the total-energy (internal + relative kinetic) conservation in the scattering process, since the energy E is a scalar.

When the coupled-channel approximation is limited to one channel (N = 1), Eq. 2.2 describes the elastic scattering in channel 1. The scalar poten- tial V (r) is then real in order to make the Hamiltonian Hermitian and the evolution operator unitary, which implies flux conservation. This situation is realized experimentally when channel 1 is chosen as the entrance chan- nel, and energy is below the first threshold ∆

²

; then all particles are still in channel 1 after the collision. When increasing energy, other channels may open; moreover, the entrance channel is not necessarily channel 1 any more.

The coupled-channel approximation may then be done with N equal to the number of open channels, which leads to an N × N matrix potential V (r).

Flux conservation between the different channels imposes that V (r) be Her- mitian. In all generality, channels may be considered in the coupled-channel approximation even if they are not open, so that the dimension N of system 2.2 can be kept the same for all energies; this is the case in the following.

When many channels are open, it may become too complicated to de- scribe each of them individually. Then a phenomenological way of describing them collectively is to allow the potential to become absorptive. In the one- channel case, this means that the potential is complex rather than real, with a negative imaginary part. This type of potential is called “optical poten- tial” (Sec. 19-d of Ref. [Tay72]) because the Schr¨odinger equation is then identical to that describing the scattering of light by an absorptive medium (complex refractive index). In the many-channel case, the matrix potential is not Hermitian any more, and one speaks of “generalized optical potential”.

Let us finally mention that in all the cases enumerated above, the ex- istence of a potential V (r) can be justified theoretically [Tay72]. However, in practice, such a potential is seldom calculated from theory; it is rather adjusted to or constructed from experimental results.

2.1.3 Rotational and time-reversal invariances

All the physical interactions known up to now are rotationally invariant. Let us define the total angular momentum

J = L

ⁱ

+ I

ⁱ₁

+ I

ⁱ₂

= L

ⁱ

+ I

ⁱ

, (2.6)

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where I

ⁱ₁

, I

ⁱ₂

are the spins of the colliding particles, which may depend on the channel i, I

ⁱ

is their sum, and L

ⁱ

is the relative angular momentum of the colliding particles. Rotational invariance implies that this total angular momentum is a constant of motion, which allows the construction of a ba- sis made of common eigenvectors of both the Hamiltonian with eigenvalue E, and the total angular momentum with eigenvalues J, M . Since the total angular momentum 2.6 only depends on the spins I

ⁱ₁

, I

ⁱ₂

and on the rela- tive angular variable ˆ r, a variable separation is possible between these spin and angular variables, and the radial variable r. The component in a given channel of a basis vector with quantum numbers E, J, M then reads

X

LⁱIⁱ

ψ

^Lⁱ^Iⁱ

(E, r) Y

M^JLⁱ^Iⁱ

(I

₁ⁱ

I

₂ⁱ

r), ˆ (2.7) where Y

M^JLⁱ^Iⁱ

(I

₁ⁱ

I

₂ⁱ

ˆ r) is an eigenvector of the total angular momentum, con- taining spin states and spherical harmonics Y

_m^Lⁱ

(ˆ r). The possible values of L

ⁱ

and I

ⁱ

depend on J and on the channel through I

₁ⁱ

and I

₂ⁱ

. One can think of Eq. 2.7 as a spinor, each component of which corresponds to a particular value of L

ⁱ

and I

ⁱ

. Let us recall that this spinor describes one channel only:

the same work has to be done for each channel.

Formally, each component of this spinor may then be considered as a

”channel” on its own, and the radial part of the basis states can then be shown (Subsecs. 15.1.1 and 16.4.1 of Ref. [New82]) to satisfy a matrix radial equation of the form

"

− d

²

dr

²

+ V (r)

#

Ψ(K, r) = K

²

Ψ(K, r), (2.8) the dimension of which depending on the number of possibles values of L

ⁱ

and I

ⁱ

in each channel. In the following, we agree to denote by N the dimension of the system 2.8 (rather than the number of genuine channels as above) and we use the term channel to designate each of these components. The channels corresponding to the same internal structure of the colliding particles then have the same threshold.

Equation 2.8 is called the radial Schr¨odinger equation, and in analogy with Eq. 2.1, the operator on the left-hand side is called the (radial) Hamil- tonian and is also denoted by H. The multiplicative matrix-operator V (r) is called the effective potential; it contains a centrifugal term L

ⁱ

(L

ⁱ

+ 1)r

⁻²

, depending on the channel.

Finally, it can be demonstrated (Subsecs. 15.1.3 and 16.4.2 of Ref. [New82]) that time-reversal invariance imposes that V (r) be symmetric; consequently, when V (r) is Hermitian because of flux conservation, it has to be real and symmetric.

2.1.4 Asymptotic behaviour of the potential

The N × N effective-potential matrix V (r) is assumed to decrease sufficiently

fast at infinity for the particular solutions described in Sec. 2.4 to be all well

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2.1. HYPOTHESES 9 behaved. We shall not be concerned with advanced mathematical notions such as existence, unicity, and analyticity of these solutions. The potential behaves asymptotically like

V (r)

_r→∞

∼ 2ηK

r + L(L + 1)

r

²

+ O(r

⁻³

), (2.9)

where η is a diagonal matrix of Sommerfeld parameters and L is a diagonal matrix of orbital-momentum quantum numbers in the different channels.

The Sommerfeld parameters are related to the wave numbers by

η = Z

₁

Z

₂

e

²

(2K)

⁻¹

, (2.10) where Z

1

and Z

2

are diagonal matrices of charge-numbers of the two particles in the different channels, and e is the electron charge.

2.1.5 Singular potential

The potential is assumed to be bounded everywhere, except possibly near the origin where it behaves like

V (r)

_r→0

∼ ν(ν + 1)

r

²

+ O(1), (2.11)

where ν is a matrix of numbers ν

ⁱ

depending on the different channels. In the following, we shall also consider the fixed-energy problem where the numbers ν

ⁱ

are allowed to become complex. Let us notice that behaviour 2.11 excludes the Coulomb potential which has an r

⁻¹

singular term at the origin. In this work, all the Coulomb potentials are regularized at the origin. Let us however mention that the supersymmetric formalism may be extended to the pure Coulomb case without major difficulty, but that some calculations of Chap.

3 concerning behaviours at the origin would then become more complicated.

For traditional potentials, one has

ν = L (regular potentials), (2.12)

i.e. matrix ν is diagonal and each of its component ν

ⁱ

is equal to the angular- momentum quantum number L

ⁱ

of the corresponding channel. Here, we do not make this restriction because supersymmetric transformations sometimes provide potentials for which

ν 6 = L (singular potentials). (2.13) In this case, we speak of singular potentials, even if this singularity does not create more trouble than a traditional centrifugal barrier, as will be seen in the following. All the singular potentials met in this work have a diagonal matrix ν, which means that the only terms of potential V (r) which may have a singularity are the diagonal terms. Moreover, the diagonal elements ν

ⁱ

of ν are always integer and bigger than the corresponding orbital- angular-momentum quantum numbers L

ⁱ

(repulsive singularity), which can be written symbolically as

ν ≥ L (in this work). (2.14)

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Other cases (non-diagonal or non-integer ν) could also be possible, but do not occur with the supersymmetric transformations studied in the following.

2.2 Self-conjugate matrix functions

All the matrix functions encountered in the following are assumed to be constituted of linearly-independent vectors. Let us define the Wronskian of two such matrices, functions of r, as

W [Ψ(r), Φ(r)] ≡ Ψ(r) ˜ d

dr Φ(r) − d

dr Ψ(r)Φ(r), ˜ (2.15) where tilde means transposition. One has

W [Ψ(r), Φ(r)] = − W ˜ [Φ(r), Ψ(r)]. (2.16) In the one-channel case, the transposition does not play any role; more- over, the Wronskian of a function with itself (denoted below as its “self- Wronskian”) vanishes. In the coupled-channel case, this is not necessarily the case; a matrix function with vanishing self-Wronskian is called self-conjugate (see Sec. 10 of Chap. XI of Ref. [Har82]) and satisfies thus

W [Ψ(r), Ψ(r)] = 0 (Ψ(r) self-conjugate). (2.17) In the following, we shall exclusively use self-conjugate matrix functions. Let us enumerate some of their properties:

• A diagonal matrix function is always self-conjugate. In particular, in the one-channel case, all functions are self-conjugate.

• If Ψ(r) is self-conjugate, Ψ(r)C is self-conjugate, with C arbitrary con- stant matrix.

• Let Ψ(r) and Φ(r) be self-conjugate. Then Ψ(r)C + Φ(r)D is self- conjugate if and only if

CW ˜ [Ψ(r), Φ(r)]D = ˜ D W ˜ [Ψ(r), Φ(r)]C, (2.18) i.e. if ˜ CW [Ψ(r), Φ(r)]D is symmetric.

In the one-channel case, the Wronskian of two functions Ψ(r) and Φ(r) vanishes if and only if these two functions are proportional to each other; i.e. a vanishing Wronskian reveals a linear dependence between two functions. This property may be generalized to the coupled-channel case for self-conjugate solutions. Let Ψ(r) and Φ(r) be two self-conjugate solutions.

• If Φ(r) = Ψ(r)C, with C arbitrary constant matrix, the Wronskian of Ψ(r) and Φ(r) vanishes; this is the analog of the one-channel case. It means that all the columns of Φ(r) can be expressed as linear combi- nations of the columns of Ψ(r). It can be shown that the inverse is also true (see Eqs. 2.23 and 2.24) and one has finally

Φ(r) = Ψ(r)C ⇐⇒ W [Ψ(r), Φ(r)] = 0. (2.19)

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2.3. SOLUTIONS OF THE RADIAL SCHR ¨ ODINGER EQUATION 11

• When only some columns (say M columns) of Φ(r) can be expressed as linear combinations of those of Ψ(r), and the N − M others are linearly independent of those of Ψ(r), the Wronskian does not vanish: it is a singular matrix of rank N − M . The inverse can also be proved.

Consequently, the rank of the Wronskian of two self-conjugate matrix func- tions indicates how many columns of one matrix are linearly independent of the columns of the other matrix.

2.3 Solutions of the radial Schr¨ odinger equa- tion

2.3.1 Vector solutions, matrix solutions

The Schr¨odinger equation 2.8 at a given wave number K is a matrix equation, so that its solutions are column vectors with N components. Since it is a second-order differential equation, homogeneous and linear, the most general vector solution depends on 2N arbitrary constants, i.e. the dimension of the vector-solution functional space is 2N .

To unify the formalism of the multi-channel case with that of the one- channel case, it is convenient to group N vector solutions to form an N × N matrix solution, the columns of which are vector solutions. The most general matrix solution then depends on N × 2N = 2N

²

arbitrary constants. This can be reformulated in the following way: let Ψ(K, r) and Φ(K, r) be two matrix solutions of Eq. 2.8, the columns of which are 2N linearly-independent vector solutions; then, using the linearity of Eq. 2.8, the most general matrix solution can be written as a generalized linear combination of Ψ and Φ,

Ψ(K, r)C + Φ(K, r)D, (2.20)

where C and D are arbitrary constant N × N matrices. Let us notice that the matrix products have to be performed on the right in order to provide linear combinations of the columns of Ψ(K, r) and Φ(K, r). Here again, we find that the most general matrix solution depends on 2 × N

²

arbitrary constants. Equation 2.20 is the precise analog of the one-channel case, where the general solution depends on two arbitrary constants only.

2.3.2 Wronskian of two matrix solutions

Let us now consider the Wronskian of two matrix solutions of the Sch¨odinger equation 2.8 at energies E and E , corresponding to matrix wave numbers K and K respectively. It satisfies

d

dr W [Ψ(K, r), Φ( K , r)] = ˜ Ψ(K, r) d

²

dr

²

Φ( K , r) − d

²

dr

²

Ψ(K, r)Φ( ˜ K , r)

= ˜ Ψ(K, r)(K

²

− K

²

)Φ( K , r)

= (E − E ) ˜ Ψ(K, r)2µΦ( K , r), (2.21)

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where the symmetry of V (r) has been used, as well as Eq. 2.5. Let us recall that E and E are scalars, while K, K and µ are diagonal matrices. For different energies E and E , this implies that the Wronskian can be obtained by integration of ˜ Ψ(K, r)2µΦ( K , r).

Equation 2.21 implies that the Wronskian of two matrix solutions at the same energy is a constant matrix,

W [Ψ(K, r), Φ(K, r)] = D. (2.22)

This relation simplifies the Wronskian calculation since it suffices to calculate it at one given r only. Moreover, it makes the check of the self-conjugation of a matrix solution easier, and allows us to determine whether two matrix solutions Ψ(K, r) and Φ(K, r) are sufficient to construct the most general solution of the equation with expression 2.20: if D is of rank N , Ψ(K, r) and Φ(K, r) are sufficient since they contain 2N linearly-independent vectors; if D is of rank N − M, M additional vector solutions are necessary.

2.3.3 Most general solution of the Schr¨ odinger equa- tion

Equation 2.22 also allows the construction of the most general solution Φ(K, r) of the Schr¨odinger equation at wave number K from one given self-conjugate solution Ψ(K, r). Actually, the general solution of the homogeneous equation

W [Ψ(K, r), Φ(K, r)] = 0 (2.23)

is

Φ(K, r) = Ψ(K, r)C, (2.24)

where the self-conjugate character of Ψ(K, r) has been used. To find the general solutions of Eq. 2.22, we use the constant-variation technique, which provides

Φ(K, r) = Ψ(K, r)C +

·

Ψ(K, r)

Z r

r⁰

Ψ(K, u)

⁻¹

Ψ(K, u) ˜

⁻¹

du

¸

D. (2.25) As expected, this general solution depends on 2N

²

arbitrary constants (the additional constant r

0

is not independent of the constants appearing in C and D). One can verify that Ψ(K, r)

^R_r^r0

Ψ(K, u)

⁻¹

Ψ(K, u) ˜

⁻¹

du is a self-conjugate solution satisfying

W

·

Ψ(K, r), Ψ(K, r)

Z _r

r⁰

Ψ(K, u)

⁻¹

Ψ(K, u) ˜

⁻¹

du

¸

= 1. (2.26)

Consequently, using Eq. 2.18, the general solution Φ(K, r) is self-conjugate if and only if

CD ˜ = ˜ DC. (2.27)

This equation introduces N (N − 1)/2 constraints on the 2N

²

arbitrary con-

stants of C and D, which means that the most general self-conjugate solu-

tion of Eq. 2.8 depends on 2N

²

− N (N − 1)/2 = N (3N + 1)/2 arbitrary

parameters. In the one-channel case, one finds 2 arbitrary parameters, which

confirms that all the solutions are self-conjugate.

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2.4. MATHEMATICAL SOLUTIONS 13

2.4 Mathematical solutions

Various definitions exist in the literature concerning particular solutions of Eq. 2.8 satisfying important analytical properties. The definitions adopted here are close to those used in Ref. [CS77]. However, they generalize these in various respects:

• As seen in Eq. 2.9, we consider the long-range Coulomb potential, which complicates the asymptotic behaviour of the wave functions.

• We consider the many-channel case in all generality, whereas it is only treated for S waves (L = 0) in Sec. IX.4 of Ref. [CS77].

• We allow the potential to be singular, according to Eq. 2.11.

In Appendix A, we establish the precise link between our definitions and the definitions used in Refs. [Tay72] and [New82]. Moreover, we correct some misprints in the latter reference.

2.4.1 Jost solutions

For any complex K, let us define the Jost matrix solution of Eq. 2.8 by its asymptotic behaviour

F (K, r)

_r→∞

∼ exp

·

ıν π

2 + ıKr − ıη ln( − 2ıKr)

¸

. (2.28)

The validity of this behaviour can be checked by direct replacement of Eq.

2.28 in Eq. 2.8, taking Eq. 2.9 into account. The Jost solution is diagonal at infinity; for = K > 0, it vanishes asymptotically, while for = K < 0, it increases exponentially at infinity.

Using the results of Sec. 2.3, it can be checked that the Jost solution is self-conjugate: since it is a solution of the Schr¨odinger equation, its self- Wronskian is constant; and since its asymptotic behaviour is diagonal, its self-Wronskian vanishes at infinity. Hence, its self-Wronskian vanishes every- where, and the Jost solution is self-conjugate.

The Schr¨odinger equation 2.8 is even in K; hence, another Jost solution can be defined in − K. It satisfies the same equation as F (K, r), and its asymptotic behaviour is still given by Eq. 2.28, but with a change of sign in K. In order to keep the argument of the complex numbers in the logarithm between − π and π, we choose a cut along the negative imaginary K axis, and use the convention

− ıK = exp( − ı π

2 )K = exp( − ı²π)ıK (² = sgn < K ), (2.29) for any complex K.

Since both F (K, r) and F ( − K, r) are solutions of the Schr¨odinger equa- tion at the same energy, their Wronskian is constant. It can be calculated from the asymptotic behaviour 2.28 and is

W [F (K, r), F ( − K, r)] = ( − 1)

^ν+1

2ıK exp( − ²ηπ), (2.30)

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where Eq. 2.29 has been used, and where an r

⁻¹

-term has vanished because of the r → ∞ limit. The term ( − 1)

^ν+1

is a diagonal matrix and has to be calculated by a series expansion. This Wronskian is a constant matrix of rank N , which means that F (K, r) and F ( − K, r) are sufficient to construct all the solutions of the Schr¨odinger equation with a formula of the form 2.20.

2.4.2 Regular solution

The regular solution Φ(K, r) is the solution of Eq. 2.8 satisfying Φ(K, r)

_r→0

∼ r

^ν+1

2

^ν

Γ(ν + 1)[Γ(2ν + 2)]

⁻¹

= r

^ν+1

[(2ν + 1)!!]

⁻¹

(ν

ⁱ

natural), (2.31) where for instance Γ(ν + 1) is a diagonal matrix, the elements of which are Γ(ν

ⁱ

+ 1). Here again, the validity of this behaviour can be verified by replacement in the Schr¨odinger equation taking Eq. 2.11 into account. Other solutions are irregular at the origin and behave like r

^−ν

there. For instance, Jost solutions are in general irregular and are often also called “irregular solutions”. Since condition 2.31 does not depend on K, the regular solution is even in K , as is the Schr¨odinger equation. Moreover, this solution is self- conjugate since its (constant) self-Wronskian vanishes at the origin, according to Eq. 2.31.

2.4.3 Jost matrix

Let us write the regular solution as a linear combination (in the sense of Eq.

2.20, with matrix multiplication on the right) of the Jost solutions Φ(K, r) = ı

2

h

F ( − K, r)K

^−ν−1

exp(²ηπ)F (K)

+F (K, r)( − K)

^−ν−1

exp(²ηπ)F ( − K)

ⁱ

, (2.32) where the factors ı/2 and exp(²ηπ) are introduced for the sake of simplicity of later formulae. One verifies that Φ(K, r) is even in K. Equation 2.32 defines the Jost matrix F (K ), which is dimensionless

⁴

. Since F (K, r), F ( − K, r) and Φ(K, r) are self-conjugate, Eq. 2.18 imposes that

F ˜ ( − K) exp(²ηπ)K

^−2ν−1

F (K) (2.33) be symmetric.

Using Eqs. 2.30 and 2.32, the Jost matrix can also be written as

F (K) = ( − K)

^ν

W [F (K, r), Φ(K, r)]. (2.34) Since this Wronskian is constant with respect to r, it can be calculated at the origin, which leads to

F (K) = ( − K)

^ν

lim

r→0

h

F ˜ (K, r)r

^νⁱ

2

^ν

Γ(ν + 1)[Γ(2ν + 1)]

⁻¹

= ( − K)

^ν

lim

r→0

h

F ˜ (K, r)r

^νⁱ

[(2ν − 1)!!]

⁻¹

(ν

ⁱ

natural), (2.35) where it has been assumed that the Jost solution is irregular at the origin.

4From now on, we shall use the term Jostmatrixrather than Jostfunction, even in the one-channel case. The same misuse of language occurs for the collision matrix.

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2.5. PHYSICAL SOLUTIONS, COLLISION MATRIX 15

2.4.4 Symmetry properties

In the general case of a non-Hermitian matrix potential, the Jost matrix has no particular symmetry, whereas for an Hermitian potential, it satisfies

F ( − K

^∗

) = F

^∗

(K) (V Hermitian), (2.36) where the star means complex conjugation. This can be shown by noticing that Φ

^∗

(K, r) is solution of the complex conjugate of the Schr¨odinger equation 2.8 and satisfies the defining relation of the regular solution 2.31, which means that

Φ

^∗

(K, r) = Φ(K

^∗

, r) (V Hermitian). (2.37) Then the comparison of the asymptotic behaviours of these two solutions, using Eqs. 2.28 and 2.32, leads to the announced property.

2.5 Physical solutions, collision matrix

In general, the solutions constructed in Sec. 2.4 have no physical interpreta- tion; they rather are mathematical intermediaries. We shall now construct solutions from which physical properties can be directly calculated.

2.5.1 All channels are open

We first consider a real positive energy above all thresholds. All the wave numbers are then real, and we agree to choose them positive (physical re- gion). Both Jost solutions are then oscillating at infinity; asymptotically, F (K, r) and F ( − K, r) have the behaviour of outgoing and incoming wave functions respectively. However, these solutions are in general irregular at the origin and do not have a direct physical interpretation. On the other hand, the regular solution Φ(K, r) has a good behaviour both at the origin and at infinity, where it behaves like a combination of incoming and outgoing solutions. Hence, we construct from it the physical solution Ψ(K, r), regular at the origin, and behaving asymptotically like (Secs. 14-b and 20-b of Ref.

[Tay72])

Ψ(K, r)

_r→∞

∼ ı 2

½

exp

·

ıL π

2 − ıKr + ıη ln(2Kr)

¸

− exp

·

− ıL π

2 + ıKr − ıη ln(2Kr)

¸

K

⁻¹²

S(K)K

¹²

¾

, (2.38)

where S(K ) is the N × N collision matrix. Notice the modification in the

logarithmic terms with respect to Eq. 2.28; to avoid determination problems,

we use Eq. 2.38 for < K > 0 only. This solution is called physical because it

appears in the partial-wave decomposition of the stationary scattering states,

which describe the collision with separate entrance channels. This actually

corresponds to experimental situations. This physical interpretation only

holds for real energies above the highest threshold.

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Comparison of Eqs. 2.38 and 2.32 for < K > 0, taking the asymptotic be- haviour of the Jost solutions 2.28 into account, leads to the following relation between the physical solution and the regular solution

Ψ(K, r) = Φ(K, r)F

⁻¹

(K) exp( − ²η π

2 )K

^ν+1

exp

·

ı(L − ν) π 2

¸

. (2.39) For < K < 0, we define the physical solution by this relation rather than by its asymptotic behaviour 2.38, in order to avoid determination problems in the logarithmic functions. From this link between Ψ(K, r) and Φ(K, r), one deduces a relation between the collision matrix and the Jost matrix

S(K) = exp

·

ı(L − ν) π 2

¸

K

^−ν−1/2

exp(²η π

2 )F ( − K) F

⁻¹

(K ) exp( − ²η π

2 )K

^ν+1/2

exp

·

ı(L − ν) π 2

¸

. (2.40)

Here again, we assume that this equation is valid for any complex K. The collision matrix has the symmetry property

S( − K ) = ( − )

^L

S

⁻¹

(K)( − )

^L

. (2.41) On the other hand, using the symmetry of expression 2.33, one gets the symmetry of the S matrix

S(K) = ˜ S(K). (2.42)

It can thus be diagonalized by an orthogonal matrix O(K) (the angles ap- pearing in O(K ) are called the mixture parameters) and the contribution of the Coulomb potential can be separated, which gives

S(K) = O(K) exp[2ıσ(K ) + 2ıδ(K)] ˜ O(K), (2.43) where σ(K) is the Coulomb phase-shift matrix (defined in Appendix B by Eq.

B.4) and δ(K) is the eigenadditional phase-shift matrix. These two matrices are diagonal and we agree to write their elements in the order of the channels.

We shall see below that in the case of threshold differences between channels, the eigenphase shifts can be attributed to the different channels if they are required to be continuous. Consequently, their order in the diagonal matrices δ(K) and σ(K ) is well defined. Channels without threshold difference have the same mass and charge partitions (see Subsec. 2.1.3), which implies that they have the same Coulomb phase shift, and that there is no ambiguity in the order of the elements of σ(K). The situation is more complicated for matrix δ(K), a particular element of which can only be attributed to a particular channel by making the coupling (non diagonal) terms in the potential matrix V (r) continuously tend to zero. By using the definition B.5 of the Coulomb scattering matrix, by defining the eigenreflection coefficient matrix

R(K) = exp[ − 2 = δ(K )], (2.44)

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2.5. PHYSICAL SOLUTIONS, COLLISION MATRIX 17 and the eigenphase matrix

ξ(K) = < δ(K ), (2.45)

one gets the decomposition of the scattering matrix

S(K) = O(K)S

C

(K )R(K) exp[2ıξ(K)] ˜ O(K). (2.46) For a real symmetric potential, one deduces from 2.36 that

S( − K

^∗

) = ( − )

^L

S

^∗

(K)( − )

^L

(V Hermitian), (2.47) which with Eqs. 2.41 and 2.42 implies that

S

^†

(K) = S

⁻¹

(K

^∗

) (V Hermitian), (2.48) where the dagger means adjoint. For real energies above the highest thresh- old, all wave numbers are real and Eq. 2.48 implies the unitarity of the S-matrix, which could be expected from flux conservation. In this case, the mixture parameters in O(K ) are real, as well as the eigenphase shifts; hence, the eigenreflexion coefficients are 1.

2.5.2 Some channels are open

Let us come back to real energies, and consider that the energy decreases across the threshold of channel N , i.e. channel N is now closed. The N − 1 first columns of the physical solution Ψ(K, r) have the same physical meaning as before, except that the outgoing solution in channel N is now replaced by a normalizable solution. The last column has no physical meaning because its last element is a solution exponentially increasing at infinity. The physical S-matrix hence reduces to the (N − 1) × (N − 1) upper left elements of the mathematical S-matrix (which is still of dimension N ). In the case of an Hermitian potential, it can be proved (Sec. 20-b of Ref. [Tay72]) that this physical S-matrix is unitary, and that its N − 1 eigenphase shifts are the continuation of N − 1 eigenphase shifts of the N × N S-matrix. The last eigenphase shift becomes complex and has no physical meaning. This continuity of eigenphase shifts allows the establishment of a unique bijection between channels and eigenphase shifts, as mentioned above.

The same phenomenon occurs at the closure of each channel: when chan- nels 1 to M are open, the physical S-matrix has dimension M , and there are M eigenphase shifts.

2.5.3 All channels are closed

Now, all wave numbers are imaginary. We agree to take them positive imag- inary (physical sheet). When all channels are closed, the system may have bound states

⁵

. This can be understood from the asymptotic behaviour of

5In fact, bound states may occur even when some channels are open. These are called bound states embedded in the continuum (Sec. 20-c of Ref. [Tay72]). We do not discuss this possibility here; it can be considered as an accidental case of a resonance.

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the regular solution Φ(K, r), obtained by combining Eqs. 2.32 and 2.28. The Jost solution F (K, r) is exponentially decreasing at infinity, while F ( − K, r) is exponentially increasing. Consequently, if for one given wave number K

0

, one can find a constant column vector C such that

F (K

0

)C = 0, (2.49)

then Φ(K

0

, r)C is a vector solution of the Sch¨odinger equation, regular at the origin, and exponentially decreasing at infinity. Hence, this solution is normalizable and can be interpreted as a bound state of the system.

Condition 2.49 is equivalent to say that the Jost matrix calculated in K

0

is a singular matrix, i.e. that

det[F (K

0

)] = 0. (2.50)

If condition 2.49 is satisfied by only one vector C, the bound state is non degenerate. If it is satisfied by M linearly independent vectors, there are M degenerate bound states at the same energy; in this case, matrix F (K

₀

) is of rank N − M , i.e. its determinant has a zero of order M. Let us notice that the maximum degeneracy degree is the number of channels N, which makes such a degeneracy impossible in the one-channel case. From the definition 2.40 of the S-matrix in terms of Jost matrix, Eq. 2.50 implies that the S- matrix has a pole in K

0

. However, the reverse property is not always true: in the following, we shall encounter cases where the S-matrix has a pole which does not correspond to a zero of det[F (K)], but well to a pole of the Jost matrix; in such a situation, there is no bound state. Both types of S-matrix poles are represented schematically in Fig. 2.1, where the “BS” zeroes and poles correspond to a bound state (zero of the Jost-matrix determinant in the upper half K plane), while the others not (pole of the Jost matrix in the lower half K plane). This shows that the Jost matrix is more fundamental than the collision matrix, since a study of the latter only can lead to confusion about the existence or not of bound states.

For an Hermitian potential, the Hamiltonian corresponding to the radial Schr¨odinger equation 2.8 is an Hermitian operator in the space of bounded functions. Consequently, its eigenvalues are real. This means that the bound states occur for real negative energies, hence for imaginary wave numbers. In the case of non Hermitian potentials, bound states occur for complex energies and have no direct physical interpretation.

2.5.4 Levinson’s theorem

This theorem connects the number of bound states n (the degenerate bound states have to be counted several times) to the sum of the eigenphase shifts.

It has been generalized to singular potentials by Swan [Swa63]. We do not demonstrate it here but conjecture a theorem valid in the coupled-channel case, with Coulomb reference potential, with possible singularity in the po- tential, and for non Hermitian potentials. It reads

R(E = 0) = R(E = ∞ ) = 1,

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2.5. PHYSICAL SOLUTIONS, COLLISION MATRIX 19

< K < K

= K = K

F (K ) S(K )

0

^BS

∞

^BS

0

^BS

0 ∞

Figure 2.1: Zeroes (0) and poles (∞) of the determinant of a schematic Jost matrix F (K) and of the corresponding scattering matrix S(K ) in the complex K plane. Only the poles and zeroes with superscript “BS” correspond to a bound state.

Trξ(E = 0) = nπ, Trξ(E = ∞ ) = (TrL − Trν) π

2 , (2.51)

where it should be noted that the dimension of the matrices varies with energy, as the number of open channels. Moreover, the eigenphase shifts have to be made continuous, i.e. the order of the eigenvalues in the diagonalization 2.46 is not arbitrary, neither is the π determination of the phases.

In order to be comprehensive, let us mention that the second line of this theorem may be modified in the absence of Coulomb potential, for S waves (L = 0), in the presence of a zero-energy bound state (see Sec. 12-g of Ref.

[Tay72]).

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Chapter 3 Supersymmetric Transformations

In this chapter, we describe how, starting from a given Schr¨odinger system of equations of the form 2.8, new Schr¨odinger systems of the same form can be constructed by supersymmetric transformations. We assume that the solu- tions of the initial system are known (either analytically or just numerically) for any value of a continuous complex-valued diagonal-matrix parameter Θ.

In the simplest case (fixed-angular-momentum case), this parameter is just the wave-number matrix, but it can also be related to other quantities such as the angular momentum matrix (fixed-energy case). From these initial- equation solutions, we are then able to construct the solutions of the trans- formed Schr¨odinger equation for the same values of the parameter Θ. In other words, everything that is known for the initial equation is known for the transformed equation.

We first present the method in its algebraic most compact form, which allows a schematic description of the different contexts in which it appeared historically (Sec. 3.1). In Sec. 3.2, we write the method explicitly for a gen- eralized coupled-channel Schr¨odinger equation, unifying the treatment of the fixed-angular-momentum and of the fixed-energy cases. This provides the expression of the transformed potential and of its solutions in terms of the initial potential and of its solutions. It also allows comparisons with results of the literature obtained in particular cases or without the supersymmetric algebraic formalism. In Sec. 3.3, we examine in detail the way the regu- lar solution and the Jost solutions (as defined in Chap. 2) are transformed by the transformation, which leads to the modification of the Jost matrix.

According to the discussion of Subsec. 2.5.3, the Jost matrix is a very impor- tant quantity, particularly concerning bound-state properties. However, Sec.

3.3 is rather tedious to read because of the different cases which have to be treated; it can be considered as a reference section. We summarize its results in Sec. 3.4 where the modification of the scattering matrix is also provided.

Section 3.5 is devoted to the establishment of useful formulae concerning the consecutive application of two supersymmetric transformations; it allows us to make a link with existing results of the literature.

21

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3.1 Historical introduction

The technique of transformation of the Schr¨odinger equation appeared in- dependently in different contexts and at different times. We shortly present these various origins here, and schematically explain how they can be con- nected to one another. This unification has only been realized recently, around 1985.

The notations used in this introduction are valid for coupled-channel non- Hermitian cases, in order to be referred to in the following of the chapter.

Let us notice however that the technique had only been developed for single- channel self-adjoint cases up to recently.

3.1.1 Transformation operators

The principle of the transformation method was first proposed in 1882 by Darboux [Dar82a]. In this first approach, the method, denoted as “Dar- boux transformation”, is presented as a property of second-order differential equations. Let

D ¯

0

Ψ ¯

0

(Θ, r) = Θ

²

Ψ ¯

0

(Θ, r) (3.1) be such an equation, where ¯ D

₀

is a second-order differential linear operator in the variable r. The bars are introduced for convenience in the follow- ing (see Subsec. 3.2.2), and ¯ Ψ designates any solution of the equation. A Darboux transformation of this equation can be performed with a first-order differential operator A

⁻₀

satisfying the so-called intertwining relation

A

⁻₀

( ¯ D

0

− Θ

²

) = ( ¯ D

1

− Θ

²

)A

⁻₀

, (3.2) where ¯ D

₁

is another second-order differential linear operator. Actually, the solutions ¯ Ψ

1

of the new equation

D ¯

1

Ψ ¯

1

(Θ, r) = Θ

²

Ψ ¯

1

(Θ, r) (3.3) are given by

Ψ ¯

1

(Θ, r) = A

⁻₀

Ψ ¯

0

(Θ, r), (3.4) as can be seen by applying A

⁻₀

to Eq. 3.1 and by using the algebraic relation 3.2.

More general operators A

⁻₀

can be used, for instance, M -order differential operators. This generalization of the Darboux method has been proposed by Crum [Cru55]. In this case, the operators A

⁻₀

are called M -order Darboux transformation operators [BS95]. It is shown in Ref. [BS95] that such an operator can always be represented as a product of M first-order Darboux transformation operators. Hence, by limiting ourselves to first-order opera- tors, as is done in the following of this chapter, we cover all the possible cases of differential transformation operators.

Supersymmetric Transformations and the Inverse Problem in Quantum

Universit´e Libre de Bruxelles

Facult´e des Sciences appliqu´ees