-symmetric zero-gap systems, conformal field theoretical kinks,
We investigate a special class of the -symmetric
quantum models being
perfectly invisible zero-gap systems
with a unique bound state at the very edge
of continuous spectrum of scattering states.
The family includes the -regularized two particle
Calogero systems (conformal quantum mechanics models
of de Alfaro-Fubini-Furlan) and their rational extensions
whose potentials satisfy equations of the KdV hierarchy
and exhibit, particularly, a behaviour typical for
We show that the two simplest Hamiltonians
from the Calogero subfamily
determine the fluctuation spectra around the
kinks arising as traveling waves in the field-theoretical
Liouville and conformal Toda systems.
Peculiar properties of the
in the associated exotic nonlinear supersymmetry
in the unbroken or partially broken phases.
The conventional supersymmetry
is extended here to the nonlinear supersymmetry
that involves two bosonic generators composed from
of the subsystems, one of which is the central charge
of the superalgebra.
Jordan states are shown to play an essential role in the construction.
-symmetric quantum systems  have many interesting properties and attract a lot of attention, for reviews see [2, 3]. Some time ago there was revealed their close relationship with certain two-dimensional integrable quantum field theories in the context of the ODE/IM correspondence [4, 5]. On the other hand, many works were devoted to investigation of diverse aspects of the -symmetric regularizations, extensions and deformations of Calogero systems [6, 7, 8, 9, 10, 11, 12, 13]. Calogero systems, as is well known, govern the dynamics of the moving poles of rational solutions to the Korteweg-de Vries (KdV) equation [14, 15, 16]. -symmetric quantum systems also reveal peculiar properties from the perspective of supersymmetric quantum mechanics [17, 18, 19, 20, 21, 22].
Solitonic (reflectionless) and finite-gap periodic quantum systems and their rational, singular on real line, limit cases are intimately related to the KdV equation via the inverse scattering transform. The KdV equation (as well as equations of the KdV hierarchy) appears as a compatibility condition of the overdetermined system of two equations imposed on one wave function. One of them has a form of the stationary quantum Schrödinger equation with a spectral parameter. Another defines the evolution of the wave function in a time variable on which potential depends as on external parameter corresponding to an isospectral deformation of the Schrödinger equation. Such reformulation of the KdV equation, introduced by Lax, naturally explains the appearance of the inverse scattering transform under construction of soliton solutions. The Lax representation also plays a fundamental role in algebro-geometric method developed for the construction of finite-gap (finite-zone) periodic and quasi-periodic solutions . The overdertermined system of equations in the Lax representation is covariant under Darboux transformations. This covariance allows to employ the Darboux transformations as alternative method for construction of both stationary and time-dependent solutions to the KdV equation and equations of the hierarchy . Darboux transformations can also be used for construction of time-dependent self-similar rational solutions to the KdV equation [25, 26]. The corresponding potentials in the last case are of the type of potentials of the Calogero model being singular functions with a double pole on the real line. Any solution of the KdV equation satisfies the ordinary differential equation known as Novikov equation which is a higher stationary KdV equation . From the viewpoint of the associated Schrödinger operator, such higher stationary KdV equation can be reinterpreted in terms of a nontrivial Lax-Novikov integral. It is this higher order differential operator that detects all the bound states in the associated Schrödinger quantum system with a soliton potential, or the edge-states of the valence and conduction bands in the spectrum of a system with periodic finite-gap potential, and both bound and edge states in the solutions to the KdV equation with solitons in a stationary asymptotically periodic background . This operator also separates the deformed plane waves of opposite chirality in the reflectionless systems with soliton potentials, and Bloch states with opposite values of quasi-momentum inside the valence and conduction bands in finite-gap systems. The Lax-Novikov higher order differential operator also exists in any quantum system with the KdV rational potential of the Calogero type. However, in rational case it is a formally commuting with Hamiltonian operator which has a non-physical nature due to a singularity of the potential : acting on physical quantum states which nullify at the potential’s pole, it transforms them into non-physical wave functions singular at the position of the pole .
In the present paper we construct and investigate a special class of the perfectly invisible -symmetric zero-gap quantum mechanical systems related to the KdV hierarchy. They represent the -regularized two-particle Calogero systems (conformal quantum mechanics models of de Alfaro-Fubini-Furlan ) and their rational extensions, which have the unique bound state at the very edge of the continuous spectrum of scattering states, and are characterized by transmission amplitude equal to one.
The -symmetric regularization is achieved here via a simple imaginary shift of the coordinate, , 111 For -symmetric deformations of the KdV and other integrable systems in spirit of , see refs. [30, 31] and [32, 33].. As a result, the Lax-Novikov operator will transform into a true integral of the corresponding quantum system. The real and imaginary parts of the potential with ‘reconstructed’ time-dependence will provide us with interesting solutions to the KdV and higher equations of the hierarchy which exhibit, particularly, the behaviour typical for the extreme (rogue) waves. The corresponding potentials are constructed via Darboux transformations which use as the seed states a complex non-physical eigenstate of zero energy of a free quantum particle and the Jordan states related to it. As a consequence, each of the obtained quantum systems will possess a unique bound state of zero energy at the very edge of the continuous part of the spectrum. Since the quantum -symmetric systems are generated from the free particle, and their potentials are non-singular on the real line, the obtained systems will be reflectionless. In the case of usual reflectionless systems with soliton potential, the transmission amplitude is a pure phase which is a rational function of energy of scattering states with zeroes and poles defined by energies of the bound states. In the the quantum -symmetric systems we study here, there is a unique bound state of zero energy that is located at the very edge of the continuous part of the spectrum. As a consequence, the transmission amplitude does not depend on energy of the scattering states and reduces to a constant value equal to . This means that the systems we consider are not just reflectionless, but are perfectly invisible systems. Such quantum systems are studied, particularly, in the context of quantum optics [34, 35, 36]. The described peculiarities of the perfectly invisible zero-gap quantum systems lead to unusual properties of the corresponding supersymmetrically extended systems. Because of the presence of the nontrivial Lax-Novikov integrals of motion, instead of the conventional supersymmetry, the Darboux-related quantum pairs will be described by exotic nonlinear supersymmetry. The anti-commutators of linear and higher order supercharges appearing here generate the Lax-Novikov integral of the extended system, which plays a role of the bosonic central charge of the superalgebra. We also will show that the two simplest Hamiltonians from the Calogero subfamily determine the fluctuation spectra around the -regularized kinks arising as traveling waves in the field-theoretical Liouville and conformal Toda systems.
The paper is organized as follows. In the next Section 2 we construct the indicated class of the quantum systems by applying the appropriate Darboux-Crum transformations to a free particle. We investigate their relationship with the stationary and non-stationary equations of the KdV hierarchy, and describe the properties of the higher-derivative Lax-Novikov integrals in such systems. We also consider there some particular time-dependent -symmetric potentials whose real and imaginary parts have a soliton-like form with a behaviour typical for extreme waves. In Section 3 we study -dimensional conformal field theoretical kinks appearing in the Liouville and conformal Toda systems and establish their relation with two simplest cases of perfectly invisible -symmetric Calogero systems. In Section 4 we discuss exotic nonlinear supersymmetry of the extended systems composed from the pairs of -symmetric zero-gap systems related by the first order Darboux transformations. Section 5 is devoted to the summary, discussion and outlook. In Appendix we briefly discuss a quantum scattering problem on the half-line.
2 Perfectly invisible -symmetric zero-gap systems
2.1 -regularized Calogero systems
Consider a quantum free particle on described by the Hamiltonian . Eigenstates of are the plane waves , , and any positive energy value is doubly degenerate. A non-degenerate eigenstate of zero energy corresponds to a limit case of the plane waves . Like , is a bounded function on . A linear independent from eigenstate of of zero eigenvalue is , that is a non-physical state unbounded at infinity. In general, a linear independent from solution to a stationary Schrödinger equation can be taken in the form
Normalization in (2.1) corresponds to the Wronskian value . The state also can be obtained from the odd linear combination of the plane wave states of energy in a limit : .
The Hamiltonian operator has an integral of motion , . It distinguishes the left- and right-moving plane wave states inside the energy doublets, . The state constitutes the kernel of , , while non-physical zero energy eigenstate is transformed by into the eigenstate , . As , the state is a Jordan state of of order .
Within a framework of Darboux transformations, one can construct the annihilator of the non-physical zero energy state : , . The conjugate operator is , and these two operators provide a factoirzation of :
Their permuted product generates a conformal quantum mechanics model  given by a Hamiltonian for relative motion of a 2-particle Calogero system
with a value of the coupling constant. Unlike , Hamiltonian is singular at . If is considered on a whole real line punctured at the origin, the potential barrier at is not penetrable and the states with supports on and do not mix dynamically. The singularity of operators , and is associated with a node at of the non-physical zero energy eigenstate of .
It can be obtained from the one-soliton KdV solution
by fixing the soliton centre coordinate to be pure imaginary and equal , and then by taking a limit . The limit eliminates a length scale which controls the size of the one-soliton solution correlated with its velocity. As a result, the stationary Schrödinger equation is invariant under the transformation , , that corresponds to a scale invariance of the system (2.3) similarly to the case of the free particle. A time-dependent solution to the KdV equation is obtained from the stationary solution by employing the invariance of (2.4) under Galilean transformations, . It is a singular solution with a moving along second order pole whose velocity is correlated with its background asymptotic value . The time-dependent solution also can be obtained directly from the one-soliton solution (2.5) in a more tricky way without making use of the Galilean symmetry. For this, after fixing , in the resulting intermediate singular one-soliton solution to the KdV equation we take a limit by preserving quadratic in terms but leaving the composed argument untouched. In such a way we obtain . Denoting then , we arrive at the same time-dependent singular solution .
Let us return to the Hamiltonian operators (2.2) and (2.3). From their factorization properties it follows that the operators and intertwine and : , . As a consequence, non-singular at physical states of on half-line are obtained from the odd linear combination of the plane wave eigenfunctions of , . They satisfy the relations and . The operator
being a Darboux-dressed free particle integral , commutes with . This is a Lax-Novikov operator for the two-particle Calogero system (2.3). It is not, however, a true integral of motion of since acting on physical eigenstates it transforms them into non-physical eigenstates of of the same energy which are singular at : .
The singularity at of the intertwining operators and and consequently of the Hamiltonian and operator can be removed by a ‘-regularization’. This is achieved by taking a complex linear combination of non-physical and physical zero energy eigenstates of ,
where is a nonzero real constant which, for definiteness, will be taken positive, . With the help of this state we construct the first order differential operators
whose kernels are, respectively, and . As before we have . But now the partner Hamiltonian operator is non-singular on the real line ,
and the first order differential operators (2.8) intertwine and ,
Hamiltonian (2.9) with a complex potential
is -symmetric, . Here is a space reflection operator, , , and a complex conjugation operator is defined by , , where is an arbitrary complex number. If we extend the definition of time reflection operator by a requirement , the time-dependent KdV equation (2.4) will be invariant under the transformation if is -symmetric : . Then the change accompanied by a shift for transforms potential (2.11) being stationary solution of the KdV equation into -symmetric time-dependent travelling wave solution of the same equation.
The bounded on real line eigenstates , , , can be considered as physical states of the -symmetric system (2.9). They are the Darboux-transformed plane wave eigenstates of that are obtained by applying to them the intertwining operator defined in (2.8). Unlike , the system (2.9) has a gapless bound state of zero energy that lies at the very edge of the doubly degenerate continuous part of the spectrum. This square-integrable on state can be obtained from the bounded but not square-integrable singlet zero energy eigenstate of by applying to the latter the intertwining operator : , . The ground state corresponds to a limit of the physical states from the continuous part of the spectrum, . It also can be obtained from the bound eigenstate of eigenavalue of the reflectionless Pöschl-Teller system given by potential (2.5) at after fixing and taking a limit . In this limit the energy gap separating the bound state from the continuous part of the spectrum disappears, and the bound eigenstate of the Pöschl-Teller system with one-soliton potential (2.5) transforms into the bound state of the system .
The peculiarity of the reflectionless -symmetric system (2.9) in comparison with Hermitian reflectionless systems is that the square-integrable on singlet state here is not separated by a gap from the continuous doubly degenerate part of the spectrum with . In this aspect it is similar to the periodic -symmetric finite-gap systems considered in . It is also worth to emphasize here that the translational non-invariance of the -symmetric system (2.9) generated from the original translation-invariant free particle system is rooted in translational non-invariance of the zero energy eigenstate (2.7) of underlying the Darboux transformation.
Unlike (2.3), the -symmetric system (2.9) possesses a true integral of motion being a Darboux-dressed integral of the free particle, , . This integral distinguishes the doublet states in the continuous part of the spectrum, , and annihilates the ground state , . The complete kernel of the third order differential operator is
It includes non-physical (undbounded) states and . These two states are not eigenstates of , but they belong to the kernels of the operators and , respectively:
A partner of the zero energy ground state of the system given by relation (2.1) is an unbounded state , . It does not belong to the kernel (2.12) like the non-physical zero energy eigenstate of does not belong to the kernel of the integral . The state is obtainable from the appropriate linear combination of the doublet states from the continuous part of the spectrum of in a limit :
The operator transforms this non-physical zero energy state into a physical ground state, , that can be compared with a similar picture in the case of the free particle system.
With respect to the Darboux transformation generator , the pre-images of the states from the kernel of are , and : , , . Unlike the zero energy eigenstate of , other two states satisfy the relations and , i.e. and are the Jordan states of of orders and , respectively. The nature of these three states related to the system is similar to that of their Darboux-counterparts (2.12) in the system , cf. Eq. (2.13).
The family of the systems
characterized by the integer parameter is a hierarchy of the -symmetric reflectionless systems that includes the free particle and the system (2.9) as particular cases corresponding to the values and . For neighbour members of the hierarchy, there are the factorization,
and the intertwining,
relations. Here the first order operators
are generated by zero energy non-physical eigenstates of , . The bounded but not square-integrable physical eigenstates of from the continuous part of the spectrum are obtained by applying to the corresponding eigenstates of of the same nature, , . These states can be constructed iteratively from the plane wave eigenstates of the free particle model,
Here we define differential operators of order :
The square-integrable zero energy ground state of is obtained from a singlet zero energy ground state of : .
The operators (2.21) intertwine directly with ,
and correspond to the Darboux-Crum transformation of order of the free particle based on non-physical eigenstate of and its certain Jordan states, see below. They allow us to construct a nontrivial integral of the system in the form of the dressed momentum operator of the free particle,
Being differential operator of order , it distinguishes the left- and right-moving eigenstates (2.20) of ,
and satisfies the Burchnall-Chaundy, supersymmetry-type relation
Its kernel of dimension is
Except the zero energy ground state , all other states from the kernel are not eigenstates of but satisfy the relations
Each of the states with is the Jordan state of of the corresponding order .
The partner (2.1) of zero energy ground state of is a non-physical state . It does not belong to the kernel (2.26), but the action of transforms it into the physical ground state . It can be obtained from a linear combination of the physical eigenstates from the continuous part of the spectrum of :
where is a constant coefficient. It also can be obtained from the plane wave solutions of the free particle by employing relation (2.20).
where is the Wronskian of the states . Here the states with are the Jordan states of of zero energy of the corresponding order : , . Such a transformation based on Jordan states corresponds to a confluent case of some Darboux-Crum transformation [41, 35, 42]. For the eigenstates of from (2.20) we have (up to a multiplicative constant factor) an alternative representation :
The case is included in (2.30), and corresponds to generation of the unique square-integrable zero energy eingenstate of from the eigenstate of . Note that the arguments in the Wronskians also can be presented in the form in terms of the plane wave eigenstates of the free particle. This corresponds to the generalized Wronskian formula for solutions of the KdV equation considered in ref.  that, in turn, can be obtained via confluent Darboux-Crum transformations .
The nature of the family of the -symmetric systems (2.16) is rather peculiar. Like any finite-gap system, each of the systems (2.16) possesses the corresponding Lax-Novikov integral of motion (2.23) that is a differential operator of order . Each of these systems is reflectionless : the plane wave states and of the free particle are transformed into the deformed plane wave eigenstates (2.20) of which propagate to the left or to the right, and are distinguished by the Lax-Novikov integral, see Eq. (2.24). The two indicated properties are also characteristic for any system with a multi-soliton potential as for their simplest one-soliton representative (2.5). Unlike the indicated conventional reflectionless systems, the -symmetric systems (2.16) are perfectly invisible. Namely, as follows from (2.21) and (2.19), the eigenstates (2.20) corresponding to the upper sign case in the asymptotic region have the form of the plane waves . In the asymptotic region these states have exactly the same form of the plane waves multiplied by the same constant factor . A similar picture is valid for solutions that correspond to the lower sign in (2.20) and have the asymptotic form in both regions and . The phase shift produced by a nontrivial potential in (2.16) is therefore equal to zero (). The transmission amplitude is equal to one, and any of these systems is indeed perfectly invisible.
The potential of is a stationary -symmetric solution of the corresponding higher order equation of the KdV hierarchy. For instance, the potential is a stationary solution to the equation
2.2 -symmetric rational extensions of the -regularized Calogero systems
The shape-invariant family (2.16) is not a unique set of perfectly invisible -symmetric systems that can be constructed by applying Darboux transformations to the free particle. A simple example of such a system of the form different from (2.16) can be obtained by taking a wave function
as a seed state for the Darboux transformation. This is a linear combination of the bound state of zero energy of the system and of its non-physical partner in the sense of (2.1), where is a constant. Requiring that is purely imaginary, the seed state (2.32) will be -invariant. Via a -odd superpotential
one can generate two super-partner potentials , where and
On the other hand, if we put in (2.33)
we obtain a complex -symmetric solution of the KdV equation (2.4), which is non-singular for . If in this time-dependent solution we put , it takes the form of the well known rational singular solution to the KdV equation (2.4) . The permitted values of the parameter indicated in (2.34) and (2.35) guarantee that the amplitude of the wave function (2.32) nowhere turns into zero. Note that the fact that the potential satisfies simultaneously the time-dependent KdV equation (2.4) and the higher order ordinary differential equation being a stationary case of (2.31) corresponds to a general property mentioned in Section 1 in relation to the Novikov equation.
The variation of the form of real and imaginary parts of the potential (2.33), (2.34) as a stationary -symmetric solution () to the higher order KdV equation (2.31) are illustrated by Figures 1 and 2 for various values of the parameters and .
The evolution of this potential in the time-dependent case (2.35) as a -symmetric solution to the KdV equation (2.4) is shown on Figure 3. It is interesting to note that near the critical value of the parameter , the graph of real part of the potential demonstrates a singular -function type behaviour while the imaginary part undergoes a flip and has a typical form. This critical value is at the very lower edge of the infinite interval (2.35) of the permitted values for the parameter in the time-dependent case. As a consequence, for the values of the parameter close to the critical value , the inverted potential reveals a behaviour typical for rogue (extreme) waves that is seen already from Fig. 3.
On the base of the function (2.32), one can construct the first order differential operators
which factorize the pair of Hamiltonians (2.9) and : , , and intertwine them : , . In correspondence with these relations, the bounded eigenstates of the system are constructed from the plane wave eigenstates of the free particle system, , where the operator is given by Eq. (2.8). The unique quadratically integrable eigenstate of zero eigenvalue of the system corresponds to the limit case of the bounded eigenstates . It is generated by applying the second order operator to the bounded eigenstate of the free particle , or, equivalently, by applying the first order operator to the ground state of zero energy of . This state is annihilated by the Lax-Novikov integral
, that satisfies the Burchnall-Chaundy relation , and distinguishes the bounded eigenstates : . The potential (2.33) can be produced directly from the free particle system via the relation
that corresponds to the second order Darboux-Crum transformation. Here the first argument of the Wronskian is the non-physical eigenstate of zero eigenvalue of , whereas the second argument corresponds to a linear combination of the eigenstate and the Jordan state of the second order, .
As yet another example of the -symmetric perfectly invisible system we present here the system described by the potential
where is, again, a purely imaginary parameter. This potential can be produced from the system by constructing the Darboux transformation based on the state
that is a linear combination of the quadratically integrable zero energy eigenstate of and of its non-physical partner . Equivalently, the perfectly transparent potential (2.39) can be produced by the relation of the form (2.38) with the Wronskian changed for , where in the last argument corresponds to the Jordan state of of the third order : . The potential (2.39) is a stationary -symmetric solution to the equation (2.31), which is non-singular provided the following restriction on the values of the parameter is introduced :
under which the wave function (2.40) nowhere turns into zero on real line .
in (2.39) transforms the -symmetric stationary solution of (2.31) into the -symmetric function that is a time-dependent -symmetric solution of the same equation (2.31) to be non-singular for all values of . In the case , potential (2.39) reduces to , that is a potential of the -symmetric perfectly transparent system (2.16) with .
The time dependence (2.35) and (2.42) for the corresponding potentials considered here as well as in a general case can be fixed by exploiting the covariance under Darboux transfromations of the Lax representation for the KdV equation and for higher equations of the hierarchy mentioned in Section 1.
The form of potential (2.39) for the stationary and time-dependent cases is illustrated by Figures 4, 5 and 6. Note that unlike the case corresponding to the potential (2.33), the critical value here cannot be approached by the time-dependent solutions (2.42).
The corresponding Hamiltonian operator is factorized, , and we also have . Here the factorizing operators have the form similar to that in (2.36) with the generating function changed for , and with the superpotential replaced by
The perfectly transparent -symmetric system is characterized by the Lax-Novikov integral