Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems
Abstract
We investigate a peculiar supersymmetry of the pairs of reflectionless quantum mechanical systems described by soliton potentials of a general form that depends on scaling and translation parameters. We show that if all the discrete energy levels of the subsystems are different, the superalgebra, being insensitive to translation parameters, is generated by two supercharges of differential order , two supercharges of order , and two bosonic integrals of order composed from Lax integrals of the partners. The exotic supersymmetry undergoes a reduction when discrete energy levels of one subsystem coincide with any discrete levels of the partner, the total order of the two independent intertwining generators reduces then to , and the nonlinear superalgebraic structure acquires a dependence on relative translations. For a complete pairwise coincidence of the scaling parameters which control the energies of the bound states and the transmission scattering amplitudes, the emerging isospectrality is detected by a transmutation of one of the Lax integrals into a bosonic central charge. Within the isospectral class, we reveal a special case giving a new family of finitegap first order Bogoliubovde Gennes systems related to the AKNS integrable hierarchy.
1 Introduction
Solitons and related topologically nontrivial objects such as kinks, instantons, vortices, monopoles and domain walls play an important role in diverse areas of physics, engineering and biology [1, 2, 3]. Darboux and Bäcklund transformations, with their origin in the theory of the linear SturmLiouville problem and classical differential geometry, proved to be very effective in their study [4, 5]. Darboux transformations [4], on the other hand, underlie the construction of supersymmetric quantum mechanics [6, 7]. Via the Bogomolny bound and the associated first order BogomolnyPrasadSommerfield equations [8, 9], supersymmetry, in turn, turns out to be closely related with the topological solitons [10, 11, 12].
Solitons and their periodic analogs appear as solutions of classical nonlinear integrable field equations, and by means of Lax representation [13] are related with reflectionless and periodic finitegap quantum systems [14, 15]. As both families of quantum systems are characterized by nontrivial, higher derivative integrals of motion, one could expect that supersymmetric extensions of them should possess some peculiar properties. This is indeed the case [16, 17, 18, 19, 20, 21], and exotic supersymmetric structures of reflectionless and finitegap systems found recently some interesting physical applications [22, 23, 24, 25, 26].
The most known example of reflectionless systems is given by a hierarchy of PöschlTeller potentials. The Schrödinger Hamiltonian with one, two, and, in general, bound states PöschlTeller reflectionless potentials controls, particularly, the stability of kinks in sineGordon, and other exotic (1+1)dimensional field theory models [1, 3, 27, 28, 29, 30, 31, 32]. These systems also appear in GrossNeveu model [33, 34]. The indicated hierarchy represents, however, only a very restricted case of a general family of soliton potentials. The latter corresponds to parametric solutions of the Kortewegde Vries (KdV) equation [2, 4, 35].
More explicitly, the Schrödinger operator is at the heart of the inverse scattering transform method of solving the classical KdV equation, for which the reflectionless potentials provide the particlelike, soliton solutions. On the other hand, the Schrödinger Hamiltonians with reflectionless potentials control the stability of the above mentioned kink solutions in dimensional field theories, and their certain supersymmetric quantum mechanical structure proved particularly to be very useful in the computing of the kink mass quantum shifts, see ref. [36].
In the present paper we study the exotic supersymmetry that appears in the pairs of reflectionless systems described by soliton potentials of the most general form. Namely, we investigate a peculiar supersymmetric quantum mechanical structure of the class of onedimensional systems described by a matrix Hamiltonian
(1.1) 
with
(1.2) 
to be soliton solutions of the KdV equation, each depending on the sets of scaling parameters, denoted here as and , and translation parameters, and . One of the possible (but not unique, see below) physical interpretations of the system (1.1), (1.2) is that it can be considered as a Hamiltonian of nonrelativistic spin particle with spindependent forces of a special form (not inducing spin flips).
A nonsoliton system of a general form (1.1), with arbitrary chosen potentials and , has just a trivial integral given by the diagonal Pauli matrix . For a special choice of potentials , this trivial symmetry is extended for supersymmetric structure related to nontrivial additional integrals of motion , . They generate a linear in , Lie superalgebraic structure , , , with the integral playing a role of the grading operator, , . It is such a linear superalgebraic structure that appears, particularly, in the Landau problem for nonrelativistic electron, where superpotential is a linear function , and (1.1) takes a form of the superoscillator Hamiltonian, see [7]. The existence of the linear supersymmetric structure is equivalent to the condition that the upper and lower components of the matrix Hamiltonian, , are related by the Darboux intertwining generators, , , being the first order differential operators and . With this observation, the construction can be generalized to nonlinear supersymmetry if the potentials and are such that the corresponding partner Hamiltonians are connected by the intertwining relations of the same form, but with and to be differentials operators of order . If this happens, the system possesses nilpotent supercharges and , , , where . They generate a nonlinear supersymmetry of the form , where is an order polynomial. The simplest example of a system with nonlinear supersymmetry is provided by a generalized superoscillator system , for which , are the usual creationannihilation bosonic oscillator operators, and the order polynomial is , see ref. [37].
The peculiarity of the system (1.1), (1.2) we study here is that the soliton potentials (1.2) are reflectionless. By a known construction based on CrumDarboux transformations, such potentials can be obtained from a free particle system, which possesses a momentum integral . It will be shown that, as a consequence, the soliton extended system is described by an exotic supersymmetric structure that includes not only one but two pairs of odd (antidiagonal) matrix supercharges, and two even (diagonal) additional nontrivial bosonic integrals being differential operators of order . The supercharges in general case are higher order matrix differential operators, two of which are of the even order , and other two supercharges are of the odd order such that . Corresponding superalgebra generated by four supercharges is nonlinear, and includes in its structure those additional nontrivial bosonic integrals of motion which are nothing else as a CrumDarboux dressed form of the free particle momentum operator. The supercharges also have a nature of the dressed integrals of motion of the free spin particle described by the Hamiltonian (1.1) with . We shall show that such a peculiar supersymmetric structure of the extended soliton systems experiences radical changes in dependence on relation between the two sets of the scaling and translation parameters of the partner potentials: the differential order of supercharges can change, and in the completely isospectral case when , one of the additional bosonic integrals transforms into the central charge of the corresponding nonlinear superalgebra. Analyzing different faces of supersymmetry restructuring, we detect, particularly, a special family of supersymmetric soliton partner potentials when one pair of supercharges reduces to the matrix first order differential operators. These first order supercharges and form between themselves a linear superalgebra corresponding to the broken supersymmetry. In such a case, one of the first order supercharges can be reinterpreted as a first order Hamiltonian of a Dirac particle. The reinterpretation provides us then with new kinkantikink type solutions for the GrossNeveu model by means of the first order Bogoliubovde Gennes system, in which a superpotential takes a meaning of a condensate, an order parameter, or a gap function depending on the physical context.
The paper is organized as follows. In the next Section, we review the general construction of soliton potentials with the help of CrumDarboux transformations, summarize the basic properties of the corresponding reflectionless quantum systems, and formulate precisely the problems related to supersymmetry of soliton systems (1.1), (1.2) to be studied here. Section 3 is devoted to the analysis of supersymmetry of nonisospectral pairs of reflectionless systems with different bound state energy levels given in terms of nonequal scaling parameters . In Section 4 we investigate the changes this supersymmetric structure undergoes in the isospectral case . Section 5 generalizes the results of Section 3 for the case of soliton pairs with completely broken isospectrality. To clarify the supersymmetry picture in extended systems with partially broken and exact isospectralities, we study in detail the case of in Section 6. In Section 6.1 we review the properties of the generic reflectionless systems to identify the ingredients to be important for further analysis. Then, in Section 6.2, we discuss a generalization of CrumDarboux transformations that is related to alternative factorizations of the basic CrumDarboux generators of order . The results of Sections 6.1 and 6.2 are employed in Sections 6.3 and 6.4 for analysis of supersymmetry in extended systems with partial isospectrality breaking. Finally, in Sections 6.5, 6.6 and 6.7 we investigate the most tricky case of supersymmetry in twosoliton extended systems with exact isospectrality. We do this first in Section 6.5 for a particular case of exact isospectrality with a common virtual subsystem. In Section 6.6 we investigate a generic case of exact isospectrality, within which we detect yet another, very special, particular case. The latter is studied in Section 6.7, and provides us with a new, first order finitegap system belonging to the AKNS hierarchy [38, 15]. In Section 7 we discuss how the results on partially broken and exact isospectralities are generalized for the systems (1.1), (1.2) with . In Section 8 we consider an interpretation of the system (1.1), (1.2) as a nonrelativistic spin particle with spindependent forces. We conclude the paper with discussion of the obtained results and their possible developments and applications in Section 9.
2 Family of reflectionless soliton systems
A CrumDarboux transformation of order , , applied to a quantum free particle generates a system characterized by the Hamiltonian [4]
(2.1) 
Here is a free particle Hamiltonian, and is a Wronskian of its eigenfunctions , , , ,
(2.2) 
A simple choice of in the form of the unidirectional plane waves , which are eingensolutions of , produces the Wronskian of the form , and, therefore, . If we take a linear independent set of linear combinations of left and right moving plane waves with for all , we obtain a nontrivial potential , which satisfies a higher order stationary KdV, , (Novikov) equation being a nonlinear ordinary differential equation with a linear highest derivative term [39, 40]. (2.1) belongs then to a class of finitegap, or algebrogeometric systems ^{1}^{1}1Finitegap periodic systems are given by the ItsMatveev representation of the form (2.1) but with substituted by a Riemann’s theta function [41]. If such a periodic potential is real and regular on , the spectrum of Schrödinger (Hill) operator is organized in valence and a conductance bands separated by gaps. (2.1) with reflectionless, soliton potential (2.4) can be considered then as the infinite period limit of a periodic or almost periodic finitegap system. In the indicated limit, the valence bands shrink, some of which can merge in this process, and transform into the nondegenerate discrete energy levels of the bound states of a resulting soliton potential; the semiinfinite conductance band turns into the continuous part of the spectrum of a reflectionless system. Quantum systems with periodic gap and nonperiodic soliton potentials (whose discrete energy levels and continuous spectrum are also separated by gaps) are characterized by the existence of the differential operator of order , related with a higher order Novikov equation, that commutes with a Hamiltonian, see below. A free particle can be treated in this picture as a zerogap system (of an arbitrary period), for which the corresponding first order differential operator is just the momentum integral . For the theory of finitegap and soliton systems including historical aspects, see [14, 42].. For real , the emergent ‘finitegap’ potential has, however, singularities on and does not disappear at . An appropriate choice of the free particle nonphysical eigenfunctions (corresponding to certain linear combinations of the left and right moving plane waves evaluated at imaginary momenta),
(2.3) 
of energies , gives rise to a nodeless Wronskian . A nonsingular parametric potential
(2.4) 
corresponds then to a reflectionless (Bargmann) system with nondegenerate states, separated by gaps, of which, of energies , , are the bound states, while the nondegenerate state of zero energy, , lies at the bottom of the doubly degenerate continuous spectrum with . From another perspective, reflectionless potential describes soliton solutions of the KdV equation.
Eigenstates of , , different from the physical bound states, are generated from eigenfunctions of the free particle, , ,
(2.5) 
where are given by Eq. (2.3). Physical nondegenerate bound states of with are obtained by the same prescription (2.5) under the choice for odd , and for even . The lowest nondegenerate state of the continuous part of the spectrum of corresponds to the eigenstate of .
Transmission scattering amplitudes for the continuous part of the spectrum , , of reflectionless system are defined by the scaling parameters [4],
(2.6) 
The states (2.5) have an alternative but equivalent representation, , generated by an sequence of the first order Darboux transformations,
(2.7) 
where denotes the set of parameters , and are the first order differential operators defined recursively in terms of the states (2.3) by
(2.8) 
(2.9) 
The first order operator annihilates the state , that is a nonphysical eigenstate of of eigenvalue . As inverse to (2.7), there is, up to an overall multiplicative constant, a relation
(2.10) 
The zero mode of the first order operator is . It is the ground state of of the energy .
A reflectionless soliton Hamiltonian admits two factorization representations
(2.11) 
In particular, the free particle gap Hamiltonian has an alternative representation From (2.11) there follow intertwining relations
(2.12) 
Let us take now a pair of soliton reflectionless systems,
(2.13) 
and consider the extended matrix Hamiltonian of the form (1.1) with and . Two sets of parameters are supposed to be completely different, or may partially coincide. If the two sets of the scaling parameters , , and , , do not coincide, the two subsystems have not only different spectra of bound states, but in accordance with (2.6), their transmission amplitudes are also different. If, moreover, for all , all the energy levels of bound states for two soliton reflectionless systems are different, and their transmission amplitudes are given by rational functions of with different zeroes and poles. Having in mind that the factorization relations (2.11) and the associated intertwining relations (2.12) are reformulated in terms of supersymmetric quantum mechanics construction, one can put a question:

What a supersymmetric structure is associated with reflectionless pair (2.13) in a completely nonisospectral case^{2}^{2}2 Using this term we neglect the fact that the continuous (scattering) parts of the spectra of the partner systems are the same, . characterized by inequalities for all ?
Such a kind of supersymmetry of the pairs of reflectionless systems was not investigated yet in the literature, but, instead, supersymmetry of the pairs (, ), , belonging to the same Darboux chain (2.12) is usually considered. In particular, the pairs of reflectionless PöschlTeller systems, see below, appear in the context of shapeinvariance [43, 44, 7], they also emerge in the infiniteperiod limit of finitegap periodic crystal structures [22, 24]. Supersymmetry of reflectionless PöschlTeller pairs (, ) was studied recently from the perspective of AdS/CFT holography and AharonovBohm effect [45].
A special choice of the parameteres
(2.14) 
results in two copies of the soliton potentials and , which describe two mutually shifted reflectionless PöschlTeller systems with bound states. Since the partner potentials under the choice (2.14) have exactly the same form, this corresponds to a particular case of a shapeinvariance, whose analog in the case of periodic supersymmetric systems was called by Dunne and Feinberg ‘selfisospectrality’ [17]. The exotic nonlinear supersymmetry of the simplest isospectral pair , with , was investigated and applied for the description of the kink and kinkantikink solutions of the GrossNeveu model [46, 24]. One can expect that the selfisospectral pair of reflectionless PöschlTeller systems with bound states should also be described by some not studied yet exotic nonlinear supersymmetric structure.
In a more general case of the choice , , different from (2.14), the partners with , , are completely isospectral, their bound states energies and transmission amplitudes coincide, but the potentials have different form. We then arrive at the natural questions related to that formulated above:

How the supersymmetric structure of a general, nonisospectral case detects the coincidence of some of the scaling parameters of two systems in (2.13)?

Particularly, for a partial coincidence of the bound states energy levels, does the supersymmetry distinguish the coincidence of the scaling parameters of the same level, , from that corresponding to the case when distinct levels, with , coincide?

Is the case of a complete isospectrality of the two systems, , , detected somehow by supersymmetric structure?

Does the case of selfisospectrality possess some special characteristics from the viewpoint of supersymmetry in comparison with a general case of isospectral systems with different form of potentials?
In what follows, we study a peculiar supersymmetric structure of the pair (2.13), and, particularly, respond the highlighted questions.
3 Supersymmetry of reflectionless pair with distinct scalings
We first investigate the supersymmetric structure of the extended system
(3.1) 
described by the pair of reflectionless PöschlTeller Hamiltonians and with and arbitrary displacement parameters and . This will allow us to trace how the restructuring of supersymmetry happens in the selfisospectral case , and to form a base for further analysis for , where we will restore index , omitted here to simplify notations, in the scaling and translation parameters.
The choice of a nonphysical eigenstate , , , of produces a Hamiltonian of reflectionless PöschlTeller system
(3.2) 
and first order operators and defined by Eq. (2.8). Operators and factorize the shifted for an additive constant Hamiltonians and ,
(3.3) 
and intertwine them,
(3.4) 
A degenerate pair of eigenstates in the continuous part, , , of the spectrum of is constructed from the free particle plane wave states,
(3.5) 
The lowest nondegenerate state with corresponds to a boundary case of (3.5),
(3.6) 
Another, bound nondegenerate state
(3.7) 
of energy is obtained from the partner, , of nonphysical eigenstate of , .
Based on intertwining relations (3.4) and their analog for the system , we construct the second order operator
(3.8) 
that intertwines the partner Hamiltonians of the extended system (3.1), . Taking into account that has an integral , one can obtain yet another, third order intertwining operator,
(3.9) 
, which is independent from the second order intertwiner .
Intertwining relations in the reverse direction are obtained by a change , that corresponds to a Hermitian conjugation of the corresponding relations, , , see Fig. 1a.
The free particle integral and intertwining relations (3.4) also generate a nontrivial integral for the reflectionless PöschlTeller subsystem ,
(3.10) 
and the analogous integral, , for . Integral (3.10) is a nontrivial operator of a Lax pair for stationary KdV equation in the nonperiodic case.
Here and in what follows, the odd and even order intertwining operators are denoted by and , respectively, while the odd order integrals of the corresponding reflectionless systems are denoted by ; the lower index indicates the differential order of these operators.
Integral (3.10) detects both physical nondegenerate states of by annihilating them . The third state of its kernel is a nonphysical eigenstate of of energy , which is a linear combination of the physical bound state of the same energy and of a nonphysical eigenstate of .
The extended system (3.1) has an obvious integral of motion . The intertwining relations together with integral (3.10) allow us to identify the nontrivial Hermitian integrals for the system ,
(3.11) 
(3.12) 
As , we can take the integral as a grading operator. It classifies then , , as bosonic integrals, , while the integrals (3.11) are identified as fermionic supercharges, , of the supersymmetric structure of the extended system . There are other possibilities to choose , which are based on reflection operators and classify the nontrivial integrals of the extended system in a way different from that prescribed by the choice . The alternative choices for find some interesting physical applications, see [22, 24, 46, 47], and we return to the discussion of this point in the last Section.
Operators (3.11) and (3.12) are the Darbouxdressed integrals of the extended system described by the Hamiltonian composed from two copies of the free particle Hamiltonian . The system possesses the set of matrix Hermitian integrals
(3.13) 
The Darboux dressing,
(3.14) 
We find the superalgebraic structure of the system by employing the intertwining and factorization relations (3.4) and (3.3). It is given by the following nontrivial (anti)commutation relations:
(3.15) 
(3.16) 
(3.17) 
(3.18) 
where ,
(3.19) 
, and to simplify the formulae, we omitted the index in the supercharges and bosonic integrals. Though in the final expression for in (3.19) the dependence on disappears, it is indicated here in the arguments having in mind a further generalization for the case, where this structure is substituted for the polynomial of order in Hamiltonian.
The extended reflectionless system (3.1) is described therefore by a nonlinear superalgebra generated by four fermionic supercharges, and , and by two bosonic integrals ^{3}^{3}3There are four bosonic integrals if one counts the integrals and ., . The fermionic integrals are constructed from the intertwining operators of the second and third orders, whose composition produces nontrivial third order integrals of Lax pairs of the nonisospectral subsystems. In this supersymmetric structure, Hamiltonian plays a role of the multiplicative central charge. The nonlinear superalgebra depends here on the scaling parameters and via the polynomials , and , but does not depend on the displacement parameters and .
4 Supersymmetry of the selfisospectral pair
For the isospectral extended system with , the partner potentials have the same form and are mutually displaced. This selfisospectral case is special from the viewpoint of supersymmetric structure. As follows from (3.17) and (3.19), for the integral , composed from the third order integrals of Lax pairs of superpartner subsystems, commutes with all the integrals, and so, transmutes into a bosonic central charge of the nonlinear superalgebra. We show now that the supersymmetric structure in this case undergoes even more radical changes.
For the following reduction takes place ^{4}^{4}4A reduction of the third order intertwining generators was discussed in a general form in [48], however, with giving no special attention to a peculiar supersymmetric structure we study here; see also [49]. :
(4.1) 
where
(4.2)  
(4.3) 
(4.4) 
Relation (4.1) means that for , the first order operator should be taken as a basic odd order intertwining operator instead of ,
(4.5) 
Note that in the limit , we have and , while for , and . This is coherent with the intertwining relations (3.4).
Because of (4.1), the third order integrals are reducible, and have to be changed for the first order irreducible integrals
(4.6) 
Integrals correspond, in accordance with (3.14), to the dressed form of the integrals , of the extended free particle system , . Alternatively, the first order matrix operator , or , can be considered as a first order Hamiltonian of the free Dirac particle of mass in (1+1) dimensions, while its dressed form, , can be identified as a Bogoliubovde Gennes Hamiltonian describing the kinkantikink solution in the GrossNeveu model [33]. Function , that appears in the structure of with and , has then a sense of a gap function [23].
The following relations are valid:
(4.7) 
(4.8) 
The employment of (4.7), (4.8) together with (4.5) gives nontrivial nonlinear superalgebraic relations
(4.9) 
(4.10) 
(4.11) 
which substitute nontrivial superalgebraic relations (3.15), (3.16), (3.17) and (3.18) of the general, nonisospectral case . Here we denoted , . As , the spectrum of is strictly positive, and the Lie subsuperalgebra generated by the first order supercharges corresponds to a broken supersymmetry. The commutes now with all the supercharges in accordance with the observation made at the beginning of the Section.
While the third order intertwining operator (3.9) is well defined at , and reduces to the integral of , the first order intertwining operator in the limit reduces to the operator shifted for an infinite additive constant term in dependence on which side the difference tends to zero. In this case extended Hamiltonian (3.1) reduces just to the two identical copies of the PöschlTeller Hamiltonians, . The integrals , , can be renormalized multiplying them by , and taking a limit . In such a way they are reduced to the trivial integrals , , of