Spectral properties of a coupled $N \times N$ potential model obtained with the help of a single non-conservative supersymmetric (SUSY) transformation starting from a system of $N$ radial Schrödinger equations with the zero potential and finite threshold differences between the channels are studied. The structure of the system of polynomial equations which determine the zeros of the Jost-matrix determinant is analyzed. In particular, we show that the Jost-matrix determinant has $N2^{N-1}$ zeros which may all correspond to virtual states. The number of bound states satisfies $0\leq n_b\leq N$. The maximal number of resonances is $n_r=(N-1)2^{N-2}$. A perturbation technique for a small coupling approximation is developed. A detailed study of the inverse spectral problem is given for the $2\times 2$ case.
We generalize eigen-phase preserving (EPP) supersymmetric (SUSY) transformations to $N>2$ channel Schrödinger equation with equal thresholds. It is established that EPP SUSY transformations exist only in the case of even number of channels, $N=2M$. A single EPP SUSY transformation provides an $M(M-1)+2$ parametric deformation of the matrix Hamiltonian without affecting eigen-phase shifts of the scattering matrix.
The paper presents a detailed analysis of the so-called "Double-Root Tangent-Polynomial" (DRtTP) reduction of the regular Gauss-reference (r-GRef) potential exactly solvable by hypergeometric functions. It is shown that the Dutt-Khare-Varshni (DKV) potential represents a remarkable particular case when the DRtTP r-GRef becomes expressible in terms of elementary functions. This is also true for its basic single- or double-step SUSY partners which are either exactly or conditionally exactly quantized by Heun polynomials (Hp-EQ or Hp-CEQ, respectively).
We describe a method for constructing vector coherent states for quantum supersymmetric partner Hamiltonians. The method is then applied to such partner Hamiltonians arising from a generalization of the fractional quantum Hall effect. Explicit examples are worked out.
Pairs of SUSY partner Hamiltonians are studied which are interrelated by usual (linear) or polynomial supersymmetry. Assuming the model of one of the Hamiltonians as exactly solvable with known propagator, expressions for propagators of partner models are derived. The corresponding general results are applied to "a particle in a box", the Harmonic oscillator and a free particle (i.e. to transparent potentials).
Position dependent mass systems can be described by a class of operators which include the Ben Daniel-Duke Hamiltonians. The usual methods to solve this kind of problems are, in general, either numerical or those looking for a connection with constant mass problems. In this paper we impose the existence of first-order ladder operators to fix our initial system. Then, we perform the first and second order supersymmetric transformations to generate families of Hamiltonians whose eigenfunctions are known analytically for a given mass profile.
Supersymmetric quantum mechanics is a powerful tool for generating exactly solvable potentials departing from a given initial one. If applied to the harmonic oscillator, a family of Hamiltonians ruled by polynomial Heisenberg algebras is obtained. In this paper it will be shown that the SUSY partner Hamiltonians of the harmonic oscillator can produce evolution loops. The corresponding geometric phases will be as well studied.
We study the scattering produced by a one dimensional hyperbolic step potential, which is exactly solvable and shows an unusual interest because of its asymmetric character. The analytic continuation of the scattering matrix in the momentum representation has a branch cut and an infinite number of simple poles on the negative imaginary axis which are related with the so called anti-bound states. This model does not show resonances. Using the wave functions of the anti-bound states, we obtain supersymmetric (SUSY) partners which are the series of Rosen Morse II potentials. We have computed the …
In this work the supersymmetric technique is applied to the truncated oscillator to generate Hamiltonians ruled by second and third-order polynomial Heisenberg algebras, which are connected to the Painlevé IV and Painlevé V equations respectively. The aforementioned connection is exploited to produce particular solutions to both non-linear differential equations and the Bäcklund transformations relating them.
We obtain accurate eigenvalues for two recently derived SUSY partner Hamiltonians. We improve the Rayleigh-Ritz variational method proposed by the authors and show how to apply the Riccati-Padé method to those particular partner potentials.
Quasinormal modes are the counterparts in open systems of normal modes in conservative systems; defined by outgoing-wave boundary conditions, they have complex eigenvalues. The conditions are studied for a system to have a supersymmetric(SUSY) partner with the same complex quasinormal-mode spectrum (or the same except for one eigenvalue). The discussion naturally includes total-transmission modes as well(incoming at one extreme and outgoing at the other). Several types of SUSY transformations emerge, and each is illustrated with examples, including the transformation among different Poschl-Tel…
We analyze the one dimensional scattering produced by all variations of the Pöschl-Teller potential, i.e., potential well, low and high barriers. We show that the Pöschl-Teller well and low barrier potentials have no resonance poles, but an infinite number of simple poles along the imaginary axis corresponding to bound and antibound states. A quite different situation arises on the Pöschl-Teller high barrier potential, which shows an infinite number of resonance poles and no other singularities. We have obtained the explicit form of their associated Gamow states. We have also constructed ladde…
Quantum Hamilton-Jacobi formalism is used to give a proof for Gozzi's criterion that for eigenstates of the supersymmetric partners, corresponding to same energy, the difference in the number of nodes is equal to one when supersymmetry (SUSY) is unbroken and is zero when SUSY is broken. We show that this proof is also applicable to the case, where isospectral deformation is involved.
The paper studies violation of conventional rules of SUSY quantum mechanics for the centrifugal potential V(r) within the limit-circle (LC) range. A special attention is given to transformation properties of the Titchmarsh-Weyl m-function under Darboux deformations of the reflective Kratzer oscillator: centrifugal Kepler-Coulomb (KC) potential plus a Taylor series in r. Since our analysis is based on Fulton's representation of a regular-at-infinity solution [Math. Nachr. 281, 1418 (2008)] as a superposition of two Frobenius solutions at the origin, we refer to the appropriate expressions as th…
We show that the existence of exceptional polynomials leads to the presence of non-trivial supersymmetry. The existence of these polynomials reveals several distinct isospectral potentials for the Schrödinger equation. All Schrödinger equations having Laguerre and Jacobi polynomials as their solutions, have non-trivial supersymmetric partners with corresponding exceptional polynomials as solutions.
Using the isospectral partners of the Bessel functions derived by Reyes et al., we find, on one hand, that these functions show non-typical supersymmetric (SUSY) behavior and, on the other, that the isospectral partner of the classical wave equation is equivalent to that of a damped system whose oscillations do not vanish in time, but show a non-harmonic shape.
The PT-symmetric square well problem is considered in a SUSY framework. When the coupling strength $Z$ lies below the critical value $Z_0^{\rm (crit)}$ where PT symmetry becomes spontaneously broken, we find a hierarchy of SUSY partner potentials, depicting an unbroken SUSY situation and reducing to the family of $\sec^2$-like potentials in the $Z \to 0$ limit. For $Z$ above $Z_0^{\rm (crit)}$, there is a rich diversity of SUSY hierarchies, including some with PT-symmetry breaking and some with partial PT-symmetry restoration.
A simple way to find solutions of the Painlevé IV equation is by identifying Hamiltonian systems with third-order differential ladder operators. Some of these systems can be obtained by applying supersymmetric quantum mechanics (SUSY QM) to the harmonic oscillator. In this work, we will construct families of coherent states for such subset of SUSY partner Hamiltonians which are connected with the Painlevé IV equation. First, these coherent states are built up as eigenstates of the annihilation operator, then as displaced versions of the extremal states, both involving the third-order ladder op…
In the present paper, a systematic approach is presented for solution of two-dimensional massless Dirac equation with external electrostatic potential applied. This approach is based on the new - asymmetric - form of SUSY-like intertwining relations. It allows to build a wide variety of pairs of SUSY-partner external scalar potentials. If one of them is simple enough to be solvable, its partner is also solvable although it may have a non-trivial dependency on both coordinates. Physically, this kind of problems is related to the description of graphene and some other materials with external pot…
Supersymmetric (SUSY) transformation operators corresponding to complex factorization constants are analyzed as operators acting in the Hilbert space of functions square integrable on the positive semiaxis. Obtained results are applied to Hamiltonians possessing spectral singularities which are non-Hermitian SUSY partners of selfadjoint operators. A new regularization procedure for the resolution of the identity operator in terms of continuous biorthonormal set of the non-Hermitian Hamiltonian eigenfunctions is proposed. It is also shown that the continuous spectrum eigenfunction has zero bino…
In a box of size $L$, a spatially antisymmetric square-well potential of a purely imaginary strength ${\rm i}g$ and size $l < L$ is interpreted as an initial element of the SUSY hierarchy of solvable Hamiltonians, the energies of which are all real for $g < g_c(l)$. The first partner potential is constructed in closed form and discussed.
The three-dimensional Schrödinger equation with a position-dependent (effective) mass is studied in the framework of Supersymmetrical (SUSY) Quantum Mechanics. The general solution of SUSY intertwining relations with first order supercharges is obtained without any preliminary constraints. Several forms of coefficient functions of the supercharges are investigated and analytical expressions for the mass function and partner potentials are found. As usual for SUSY Quantum Mechanics with nonsingular superpotentials, the spectra of intertwined Hamiltonians coincide up to zero modes of supercharge…
It is commonly known that the Fokker-Planck equation is exactly solvable only for some particular systems, usually with time-independent drift coefficients. To extend the class of solvable problems, we use the intertwining relations of SUSY Quantum Mechanics but in new - asymmetric - form. It turns out that this form is just useful for solution of Fokker-Planck equation. As usual, intertwining provides a partnership between two different systems both described by Fokker-Planck equation. Due to the use of an asymmetric kind of intertwining relations with a suitable ansatz, we managed to obtain …
We propose a new SUSY method for construction of the quasi-exactly solvable (QES) potentials with three known eigenstates. New QES potentials and corresponding energy levels and wave functions of the ground state and two lowest excited state are obtained. The proposed scheme allows also to construct families of exactly solvable non-singular potentials which are SUSY partners of the well-known ones.
We use the Fock-Ivanenko formalism to obtain the Dirac equation which describes the interaction of a massless 1/2-spin neutral fermion with a gravitational field around a Schwarzschild black hole (BH). We obtain approximated analytical solutions for the eigenvalues of the energy (quasi-normal frequencies) and their corresponding eigenstates (quasi-normal states). The interesting result is that all the excited states [and their supersymmetric (SUSY) partners] have a purely imaginary frequency, which can be expressed in terms of the Hawking temperature. Furthermore, as one expects for SUSY Hamil…
We review the higher-order supersymmetric quantum mechanics (H-SUSY QM), which involves differential intertwining operators of order greater than one. The iterations of first-order SUSY transformations are used to derive in a simple way the higher-order case. The second order technique is addressed directly, and through this approach unexpected possibilities for designing spectra are uncovered. The formalism is applied to the harmonic oscillator: the corresponding H-SUSY partner Hamiltonians are ruled by polynomial Heisenberg algebras which allow a straight construction of the coherent states.
The $\cal PT$-symmetric complexified Scarf II potential $V(x)= - V_1 \sech^{2}x + {\rm i} V_2 \sech x \tanh x$, $V_1>0$ , $V_{2}\neq 0$ is revisited to study the interplay among its coupling parameters. The existence of an isolated real and positive energy level that has been recently identified as a spectral singularity or zero-width resonance is here demonstrated through the behaviour of the corresponding wavefunctions and some property of the associated pseudo-norms is pointed out. We also construct four different rationally-extended supersymmetric partners to $V(x)$, which are $\cal PT$-sy…
The quasi-degeneracy between the single-particle states $(n,\,l,\,j=l+1/2)$ and $(n-1,\,l+2,\,j=l+3/2)$ indicates a special and hidden symmetry in atomic nuclei---the so-called pseudospin symmetry (PSS)---which is an important concept in both spherical and deformed nuclei. A number of phenomena in nuclear structure have been successfully interpreted directly or implicitly by this symmetry, including nuclear superdeformed configurations, identical bands, quantized alignment, pseudospin partner bands, and so on. Since the PSS was recognized as a relativistic symmetry in 1990s, there have been co…
In this paper we start with proving that the Schrödinger equation (SE) with the classical $12-6$ Lennard-Jones (L-J) potential is nonintegrable in the sense of the differential Galois theory (DGT), for any value of energy; i.e., there are no solutions in closed form for such differential equation. We study the $10-6$ potential through DGT and SUSYQM; being it one of the two partner potentials built with a superpotential of the form $w(r)\propto 1/r^5$. We also find that it is integrable in the sense of DGT for zero energy. A first analysis of the applicability and physical consequences of the …
We make a detailed study of the first and second-order SUSY partners of a one-dimensional free Hamiltonian with a singular perturbation proportional to a Dirac delta function. It is shown that the second-order transformations increase the spectral manipulation possibilities offered by the standard first-order supersymmetric quantum mechanics.
We study complex potentials and related non-diagonalizable Hamiltonians with special emphasis on formal definitions of associated functions and Jordan cells. The nonlinear SUSY for complex potentials is considered and the theorems characterizing its structure are presented. We present the class of potentials invariant under SUSY transformations for non-diagonalizable Hamiltonians and formulate several results concerning the properties of associated functions . We comment on the applicability of these results for softly non-Hermitian PT-symmetric Hamiltonians. The role of SUSY (Darboux) transfo…
Using the disconjugacy properties of the Schrödinger equation, it is possible to develop a new type of generalized SUSY QM partnership which allows to generate new solvable rational extensions for translationally shape invariant potentials having a finite bound state spectrum. For this we prolong the dispersion relation relating the energy to the quantum number out of the physical domain until a disconjugacy sector. The prolonged excited states Riccati-Schrödinger (RS) functions are used to build Darboux-Bäcklund transforms which give regular isospectral extensions of the initial potential. We…
In the framework of SUSYQM extended to deal with non-Hermitian Hamiltonians, we analyze three sets of complex potentials with real spectra, recently derived by a potential algebraic approach based upon the complex Lie algebra sl(2, C). This extends to the complex domain the well-known relationship between SUSYQM and potential algebras for Hermitian Hamiltonians, resulting from their common link with the factorization method and Darboux transformations. In the same framework, we also generate for the first time a pair of elliptic partner potentials of Weierstrass $\wp$ type, one of them being r…
We study first the supersymmetric quantum mechanics (SUSY QM), specially the cases of the harmonic and radial oscillators. Then, we obtain a new Wronskian formula for the confluent SUSY transformation and apply the SUSY QM to the inverted oscillator. After that, we present the polynomial Heisenberg algebras (PHA). We study the general systems described by PHA: for zeroth- and first-order we obtain the harmonic and radial oscillators, respectively; for second- and third-order PHA, the potential is determined in terms of solutions to Painlevé IV and V equations ($P_{IV}$ and $P_{V}$), respective…
In some recent articles we developed a new systematic approach to generate solvable rational extensions of primary translationally shape invariant potentials. In this generalized SUSY QM partnership, the DBT are built on the excited states Riccati-Schrödinger (RS) functions regularized via specific discrete symmetries of the considered potential. In the present paper, we prove that this scheme can be extended in a multistep formulation. Applying this scheme to the isotonic oscillator, we obtain new towers of regular rational extensions of this potential which are strictly isospectral to it. We…
