Non-stationary SQM/IST Correspondence and CPT / PT -invariant paired Hamiltonians on the line

We fill some of existed gaps in the correspondence between Supersymmetric Quantum Mechanics and the Inverse Scattering Transform by extending the consideration to the case of paired stationary and non-stationary Hamiltonians. We formulate the corresponding to the case Goursat problem and explicitly construct the kernel of the non-local Inverse Scattering Transform, which solves it. As a result, we find the way of constructing non-hermitian Hamiltonians from the initially hermitian ones, that leads, in the case of real-valued spectra of both potentials, to pairing of CPT / PT -invariant Hamiltonians. The relevance of our proposal to Quantum Optics and optical waveguides technology, as well as to non-linear dynamics and Black Hole Physics is briefly discussed.


Introduction
Exact solvable models have always played a distinctive role, especially in the realm of Quantum World.The use of various kinds of approximations in the course of modeling quantum phenomena is quite justified for natural materials, but it becomes a problem for designing their artificial analogues with improved or preset characteristics.Here, the exact solvability becomes a necessary theoretical tool, that allows one to predict the response of a quantum system to an external source, and, thereby, to control its behavior in different regimes of interaction with driving forces.An even more intriguing situation is realized in the case of using, on the theoretical side, techniques for generating new exactly solvable models that have principally new properties and characteristics compared to the initial ones.Such new properties can arise as due to changes in the type of interaction between the internal ingredients of a quantum system, as well as due to the specifics of its interaction with external sources and fields.A clear understanding, on the theory side, of the processes occurred in the system is an additional, important, though by no means the last, key to their successful implementation in practice.
Currently, it has known a sufficient number of techniques for generating new exact solutions from already known ones (see, for example, monographs and paper collections [1][2][3][4][5][6][7]).One of these techniques, developed since the late 40s in quantum scattering theory, is the inverse scattering method [8][9][10][11][12], which leads to a remarkable and non-standard result: restoring the potential from the known spectrum is an ambiguous task.What was treated as a bug before became a very useful feature after, when it were established new relations between exactly solvable potentials of Quantum Mechanics, new interconnections between non-linear equations and their solutions, symmetry structures behind and many more.After seminal papers on solitons and non-linear dynamics [13][14][15], the inverse scattering method was renamed into the Inverse Scattering Transform, so that we will follow this nomenclature, irrespectively of the presence or absence of non-linearity in the system.
Another profound method of generating new solutions with exactly solvable dynamics is Supersymmetric Quantum Mechanics, whose development, after the foundation in early 80s [16][17][18][19][20], revealed similarities to the inverse scattering method.In particular, (almost) the same spectrum may correspond to potentials that are completely different in shape.This observation made it possible to put studies of quantum mechanical problems on a more solid ground than before, and to establish exact solutions even in the case when conventional approximations (WKB and others) failed.(See, e.g., Refs.[21,22] in this respect.)The latter becomes extremely important in engineering based on Quantum Optics communication networks [23,24], as well as in designing quantum field-effect diodes and transistors [25,26].In both cases, the employment of the Supersymmetric Quantum Mechanics [27][28][29][30][31][32] demonstrates principle advantages of the approach.
The purpose of this paper is to fill some of existed gaps in the correspondence between Supersymmetric Quantum Mechanics (SQM) [16][17][18][19][20] and the Inverse Scattering Transform (IST) [8- where V ± (x) are the potentials for (almost) isospectral Hamiltonians H ± = −∂ 2 x + V ± (x).If we fix the initial Hamiltonian as H + , when the stationary Schrodinger equation Such a wave function is associated with the ground state of the "bosonic" Hamiltonian H + .This Hamiltonian admits the factorization by supercharges Q † and Q, which are first order differential operators with (derivative of) the superpotential W (x).Then, the "bosonic" and "fermionic" parts of the super-Hamiltonian are constructed as and the spectra of H ± are the same, except for the ground state.Now, let's factorize the Hamiltonian H − with other supercharges, Q and Q † , Taking the definition of V − (x) from eq. ( 1), we get Mielnik [48], after van Kampen [49], marked the general solution to the Riccati equation ( 6) for f (x).It comes as Here there has been appeared a constant λ, whose role will be central in what follows.
For now, we have established the correspondence between Q and Q supercharges through the "fermionic" part of the super-Hamiltonian H − , that is, Here, we have introduced which obeys Clearly, H+ ≠ H + , and H+ corresponds to the Hamiltonian with new potential But what about the spectra of these two Hamiltonians?It turns out that the Hamiltonians H + and H+ are also (almost) isospectral, via their parent Hamiltonian H − .Indeed, we can write i ⟩ are the eigenstates of H − .So that, the energies (eigenvalues of H+ ) will be given by E i , i = 1, . . ., N .However, as in the case of H + , one needs to add the ground state Q| ψ(+) 0 ⟩ = 0 [48].In the coordinate representation, the corresponding wave function Ψ0 looks as It is straightforward to verify that H+ Ψ0 = 0, hence the energy of the ground state of H+ is also Other eigenstates of two Hamiltonians H + and H+ are related to each other via so that the normalizations of wave functions {Ψ 0 , ψ i } and { Ψ0 , ψ(+) i } are different.This situation, when the eigenvalues of two spectra are equal, but the normalizations of a finite number of point spectra wave functions are different, is well-known in the IST, and it has been considered, e.g., in [35][36][37].
According to Abraham & Moses, Ref. [37], the wave functions of two isospectral Hamiltonians are related to each other via non-local transformation with the kernel of the transformation K(x, y).Both eqs.( 13) and ( 14) i ⟩, with local (eq.( 13)) and non-local (eq.( 14)) transformation operators U , respectively.
And, as it became clear after the results of [38], the kernel of the non-local transformations ( 14) should be taken to be with a constant Λ, that (see [37] for details) makes it possible to construct a new, isospectral, Hamiltonian with the potential Comparing two equations, eq. ( 16) and eq.( 11), to each other, one may conclude that K(x, x) = −ϕ(x).Hence, eq. ( 15) turns into eq.( 9) as soon as (cf.eq. ( 2)) λ = Λ/N 2 0 .In addition, we have to check the following equation on the kernel which holds for the kernel (15).Other aspects of the SQM/IST correspondence, including the general non-equivalence of the approaches, can be found, e.g., in [39][40][41][42]51].Note that for confined potentials, with discrete (point in the nomenclature of [37]) spectra, the equivalence of the Abraham-Moses to the SQM approach was confirmed in [43,46].So that we will focus just on this case.Thus, different approaches to the construction of new Hamiltonians -SQM and the IST -are inherently related.And we can use this correspondence between two approaches in both directions.
Here we have started with the SQM Hamiltonian construction, and turned to the IST technique after.We can use the reverse procedure equally.That is, we can begin with the IST approach first, to make the conclusion on the structure of the paired SQM Hamiltonian after that.

The 1D SQM/IST Correspondence: Non-stationary case
Elaborating on the construction of new isospectral Hamiltonians with time-dependent potentials, we have to sum up the previous achievements in the stationary case.First, there is the correspondence between two different approaches -the Inverse Scattering Theory and Supersymmetric Quantum Mechanics -that are used to generate a new isospectral Hamiltonian ( H+ in the case) to the initial one (H + in the used here notation).This correspondence is exact for confined potentials that includes the harmonic potential as well.
Second, the SQM/IST correspondence is realized quite differently on the both sides.As we have noticed before, the relation between eigenstates (wave functions) of H+ and H + is established either by two consequent local transformations with different supercharges (cf.eq.( 13)), or by the single non-local transformation with the integral kernel (cf.eq. ( 14)).Any of these transformations include specific combinations of the ground state wave functions, eqs.( 9) or (15), with some unspecified, and generally complex-valued, constant λ. (Recall, Λ in ( 15) is related to λ in (9) via the square of the normalization constant N 0 .)The natural requirement that restricts this constant is For the rest, this constant can be chosen freely in the domain of the normalizability of the H+ wave functions [44].We will turn to the discussion of this aspect of λ later on, after reviewing the nonstationary extension of the SQM/IST correspondence.
And third, the considered in [38] SQM/IST correspondence works in the stationary case.That is, we have the correspondence between isospectral Hamiltonians, entering the stationary Schrodinger equations.Below, we will extend this picture on two notable cases, when the correspondence is established as between non-stationary Hamiltonians, as well as between stationary and non-stationary Hamiltonians.
Let's begin with the SQM/IST correspondence for two non-stationary isospectral Hamiltonians.The time-dependent extension on the IST side and its relation to the (S)QM approach has been elaborated in [52,53].There, it was proposed the following extension of the kernel K(x, y): with constant λ.Using the developed in [37] technique (see Appendix of [37] for details on the stationary case), one can set up the Goursat problem as It is straightforward to verify that the kernel (19) satisfies the second equation of ( 20) for wave functions of the complete Schrodinger equation with H + , The new potential Ṽ+ (x, t) enters the twinned with H + Hamiltonian H+ = −∂ 2 x + Ṽ (x, t).Its wave functions ψ(+) (x, t) also obey the non-stationary Schrodinger equation, now with H+ : The SQM side of the story was developed in Refs.[52][53][54][55][56]. (See also Refs.[57][58][59][60][61][62]76] for the development and applications of methods by Bagrov et al.)The supercharge Q, associated with H + , is formed from the superpotential W (x, t), similarly to eqs.(3).Concerning the supercharge Q, it is realized in the following way (see Refs. [58,59] for the detailed analysis of the construction of [52][53][54][55][56]): The functions l(t) and ϕ(x, t) are generally complex-valued functions.Then, the intertwining relation, which is a paraphrase of the isospectrality, makes it possible to derive the relation [52,53] between the new and the old potentials, And the function ϕ(x, t) is restricted to satisfy Note also that the requirement of real-valued potentials in (25) leads to the following constraints: so that the non-triviality of l(t) requires a complex-valued ϕ(x, t).At the same time, dealing with a real-valued potential to get real-valued eigenvalues of the Hamiltonian is an excess requirement [63][64][65].We will turn to the more detailed discussion of this point in the next section.
For now let us note that, from the construction of the kernel and supercharges, it is clear that we are dealing with non-stationary paired Hamiltonians.The SQM/IST correspondence at the level of the supercharge/kernel relation is established by comparing the right hand sides of Ṽ+ (x, t) in ( 20) and (25).As a result, we obtain a relationship between the spatial derivative of the transformation kernel and the derivatives, both spatial and temporal, of the basic ingredients of the supercharge.Thus, in contrast to the stationary case, the SQM/IST correspondence is realized indirectly, through the derivatives of the main quantities.
However, the direct realization of the SQM/IST correspondence is restored within a slightly different, but important, extension of the stationary case, which will formally connect stationary and non-stationary Hamiltonians.Now, we are turning to the assembling of this construction.

The 1D SQM/IST Correspondence: Stationary to nonstationary Hamiltonians pairing
Let's carry on with a brief discussion on the kernel structure for the stationary case.Equation ( 15) can be generalized to However, what matters is the ratio of these two constants, which we have denoted as Changing the λ does not affect the spectrum, affecting the potential and wave functions, respectively.Therefore, we can instantly change the λ in its value, within its range, determined by relation (18).Or, we can relate any instant λ changing to some unique value of the time parameter.Smooth changes form a function of time λ(t) after that.Thus, we would like to investigate consequences of using the following kernel template on the IST side: where Ψ 0 (x) is the ground state wave function of H + (cf.eq. ( 2)).
With the kernel of non-local (integral) transformations of wave functions (29), we expect that the supercharges from eqs. (5) will become functions of time, via Then, the time dependence of the H+ ground state wave function, can be treated as a time-dependent geometric phase, i.e., Recall, wave functions with time-dependent geometric phase naturally appear in quantum systems with time-dependent boundary conditions.And a by-product of the consideration is the "covariantization" [78] of the non-stationary Schrodinger equation in temporal direction.
Taking the latter into account, let us formulate the considered problem in more general terms.
Lemma.Suppose that two sets of wave functions, ψ ε (x) and Ψ(x, t), are related to each other via non-local transformations The kernel of these transformations is constructed out of the stationary wave functions ψ(x, t) = e −iεt ψ ε (x), which obey the Schrodinger equation If the wave function Ψ(x, t) obeys the generalization of the Schrodinger equation, in which D t = ∂ t + A t is a "covariantization" of the time derivative by a "compensator" field A t , determined by D t K(x, y; t) = 0, then the relation between the paired potentials and the Master equation for the kernel (the Goursat problem) looks as follows: Proof.Let's act with (H − iD t ) operator on the wave function (33).This operation can be split in two parts.One of them is where we have used eq.( 35).The other part of eq. ( 36) contains By use of we finally arrive at Equating to zero the r.h.s. of the latter equation, we get the system of equations (37).■ In the notation of the paper, we have proved that the usage of the kernel (29) results in the construction of the new, paired to H + , effective Hamiltonian H+ , extended by the interacting with the "compensator" A t term.This field is fixed by the requirement of having the "covariantly constant" kernel, i.e., ∂ t K(x, y; t) Then, for K(x, y; t) (34), so that the relation between the initial and the final, after the IST, potential in our case (with the kernel ( 29)) becomes Comparing the latter expression with that of eq. ( 25), we can establish the direct correspondence between the IST and the SQM quantities: Therefore, the usage of the special IST kernel (29) makes it possible: 1) to construct a new paired time-dependent potential from the initially stationary one; 2) to realize the SQM/IST correspondence directly in the non-stationary case, with the special kernel of the IST (34).
Another interesting feature behind our proposal can be figured out from the analysis of the potential (42).The requirement to have a real-valued potential Ṽ (x, t) leads to the following relation between the kernel and the "compensator" field: Hence, the kernel K(x, y; t) (34) has to be complex-valued, to provide the non-triviality of A t (x, t).
For the IST from a stationary system to a non-stationary one, it means that the parameter λ(t) = C 1 (t)/C 2 has to be a complex-valued function of time.Then, according to (33) and (34), wave functions of the paired Hamiltonian are principally non-stationary ones, in the sense that they are nonfactorizable in time and space coordinates, and they are complex functions.If so, they correspond to scattering states, so that the IST (33) with the kernel (34) may, for instance, transform bound states to scattering states.Even more interesting situation takes place when we omit the restriction on the new scattering potential to be a real-valued function of time and space coordinates.In this case we still have a possibility to operate with Hamiltonians having real-valued energy spectra.It happens, for example, for the so-called PT -invariant Hamiltonians [64][65][66][67][68][69][70][71][72][73][74], properties of which are nicely reviewed in Refs.[75], [76] and [77].
Let's apply the PT transformations to the potential (42).Under the PT transformations, V + (x) → V + (−x).For K(x, y; t) we have (recall, C 2 is a constant) Upon the action of PT operators on the imaginary part of a complex-valued potential (42) we obtain And if C 1 (−t) = −C 1 (t), the iIm Ṽ+ (x, t) stays to be PT invariant.Therefore, the restriction on C 1 (t) to be an odd function of time provides, together with the even parity of the potential V + (x), the invariance of the paired potential under the PT transformations.And the PT invariance is not a drawback for the factorization and application of Supersymmetry to the consideration.(Cf., e.g., Refs.[68,69,[72][73][74]76] in this respect.)Clearly, the supercharge of the paired Hamiltonian, Q = l(t) (Q − ϕ(x, t)), is constructed out of the initial even-parity Hamiltonian supercharge Q, and the determined by eqs.(43) functions with C 1 (−t) = −C 1 (t).(See Appendix for details on the paired to a simple harmonic oscillator PT -invariant Hamiltonian.)Hence, in the case of real-valued energy Hamiltonians, we have the SQM/IST Correspondence between i) stationary and non-stationary Hamiltonians; ii) bound, scattering and tunnelling states; iii) CPT -invariant and PT -invariant Hamiltonians.
Early, relations between CPT /PT -invariant Hamiltonians were established merely for non-stationary systems.(See, e.g., [74,76], and Refs.therein.)Furthermore, even if all the restrictions on the potential (like the requirement of dealing with real-valued or PT -invariant potentials) will be omitted, we will take another possibility to construct paired complex-valued Hamiltonians.As it is known from classical electrodynamics, choosing specific boundary conditions leads to the appearance of quasi-normal modes with complex-valued frequencies.The complex-valued frequencies correspond to complex energies even for a real-valued potential.Therefore, the pairing between real-valued (stationary) and complex-valued (non-stationary) potentials may be realized as a result of imposing the specific boundary conditions.The quasi-normal modes play, in particular, an important role in waveguides, resonators, and in Black Hole Physics as well.So that, the proposed here mechanics of generation of new exactly-solvable potentials may also find its natural application in these domains of study.

Discussion and final remarks
In summary, we propose a new mechanism for generating paired potentials based on a very specific time-dependent extension of the IST kernel or the SQM supercharge.The standard generalization of the Abraham-Moses construction [37] to the non-stationary case [52][53][54] assumes the replacement of stationary wave-functions with non-stationary ones.At the same time, the integration constant, coming from the general solution to the Riccati equation [48,49] and entering either the IST kernel or the SQM supercharge, remains unchanged.Our proposal is based on the opposite consideration, when the integration constant becomes a function of time, while the rest of the IST kernel or the SQM supercharge remains the same as in the stationary case.So that, if the standard non-stationary generalization of the Abraham-Moses construction pairs time-dependent Hamiltonians, our approach admits a non-standard and non-trivial pairing of time-independent and time-dependent Hamiltonians.
Other non-trivial features of our proposal consist in the correspondence between real-valued and complex-valued potentials, since the elaborated here procedure of the paired Hamiltonian construction results, generally, in the appearance of a complex-valued potential after the IST employment.
If the complex-valued potential is not constrained, the eigenvalues of a complex-valued Hamiltonian are usually complex.And it can be viewed in different ways.For instance, the complex part of energy values may indicate the dissipation in real physical systems, that describes the natural energy loss, e.g., in Quantum Optics, or in optical fibers.Another way to deal with complex-valued energies is to consider effects of tunnelling through barriers suggested by the potential shape, or by specific boundary conditions.Note that the role of boundary conditions in the twinning of Hamiltonians with real-valued and complex-valued potentials becomes fundamental, since setting the specific boundary conditions is a most natural way to generate quasi-normal modes with complex-valued frequencies.Therefore, following the proposed here recipe, one may construct, on the ground of a hermitian stationary exactly-solvable Hamiltonian, the effective exactly-solvable time-dependent Hamiltonian for a system with the energy dissipation.
The requirement to have real-valued energies in the spectrum of a complex-valued Hamiltonian leads us to the PT -invariant quantum mechanics.(See, e.g., [75][76][77] and Refs.therein.)We established the set of constraints for the initial hermitian time-independent Hamiltonian potential and the kernel of the non-local IST, that leads to the time-dependent complex-valued paired potential.The standard SQM and, partially, the PT -invariant SQM gained in popularity in Solitons Theory [79,80], Optical Waveguides [27,30], models of Analogue Black Holes [81], diffusion theory [82,83] and many more.(See, for instance, Ref. [84] for a review on modern trends in SQM.)So we hope that the obtained here outcomes of our work will give an impact on investigations in the mentioned as well as in other branches of modern science and technology.
Finally, let's pay special attention to the application of our results in Black Hole Physics.We have mentioned the Quasi-Normal Modes, which, together with the phenomenon called Superradiation, play a very important role in modern studies of Black Holes.A time ago it was established the isospectrality of the Regge-Wheeler and Zerilli equations for spin-2 perturbations over a specified gravitational background [85].However recently, in [86], it was observed that this phenomenon does not properly work for a general gravitational background.It would be interesting to investigate, on the ground of the obtained by us results, the (non-)isospectrality between equations for spin-2 perturbations over stationary and non-stationary gravitational background with formally complexvalued effective potentials.We hope to report on this and other outcomes of our work in future communications.