Isospectral Partners Research Articles

Linearly stable and unstable complex soliton solutions with real energies in the Bullough-Dodd model

We investigate different types of complex soliton solutions with regard to their stability against linear perturbations. In the Bullough-Dodd scalar field theory we find linearly stable complex PT-symmetric solutions and linearly unstable solutions for which the PT-symmetry is broken. Both types of solutions have real energies. The auxiliary Sturm-Liouville eigenvalue equation in the stability analysis for the PT-symmetric solutions can be solved exactly by supersymmetrically mapping it to an isospectral partner system involving a shifted and scaled inverse cosh-squared potential. We identify exactly one shape mode in form of a bound state solution and scattering states which when used as linear perturbations leave the solutions stable. The auxiliary problem for the solutions with broken PT-symmetry involves a complex shifted and scaled inverse sin-squared potential. The corresponding bound and scattering state solutions have complex eigenvalues, such that when used as linear perturbations for the corresponding soliton solutions lead to their decay or blow up as time evolves.

Variety of double knock out barrier option for sustainable financial management

<abstract><p>Options are financial contracts that are based on an underlying security and are useful for both hedging and speculating on future market trends. New financial tools are constantly being developed for sustainable financial management. In order to define new financial instruments, the BS Hamiltonian, in conjunction with a potential function, is particularly important for modelling path-dependent options. It is demonstrated here how supersymmetry provides a natural framework for generating various options, particularly using higher order supersymmetry to find and examine numerous isospectral partners of the double knock out barrier option.</p></abstract>

Supersymmetry with self-consistent Schrödinger–Poisson equations: finding partner potentials and breaking symmetry

It is shown that isospectral Hamiltonians and partner potentials can be found for self-consistent solutions of the Schrödinger and Poisson equations in the presence of identical non-interacting electrons. Perturbation of these systems by an external electric field can be used to break symmetry and spectrally distinguish between states. For a given pair of partner potentials, symmetry may also be broken by a change of electron density or temperature.

Isospectral and square-root Cholesky photonic lattices

Cholesky factorization provides photonic lattices that are the isospectral partners or the square root of other arrays of coupled waveguides. The procedure is similar to that used in supersymmetric quantum mechanics. However, Cholesky decomposition requires initial positive definite mode coupling matrices and the resulting supersymmetry is always broken. That is, the isospectral partner has the same range than the initial mode coupling matrix. It is possible to force a decomposition where the range of the partner is reduced but the characteristic supersymmetric intertwining is lost. As an example, we construct the Cholesky isospectral partner and the square root of a waveguide necklace with cyclic symmetry. We use experimental parameters from telecommunication C-band to construct a finite element model of these Cholesky photonics lattices to good agreement with our analytic prediction.

SUSY Quantum Mechanics for PT Symmetric Systems

A new way of constructing SUSY partner potentials with PT symmetry is proposed. In this construction, PT symmetric superpotentials generate PT symmetric SUSY partners and supercharges that satisfy commutator-anti-commutator relations of close superalgebra SL(1/1). Conversely, every PT symmetric Hamiltonian having zero or non zero ground state energy can be generated using PT symmetric superpotentials. Further, superpotentials having separate P and T symmetries can generate strictly isospectral SUSY partners if the wedge of integration can accommodate two asymptotically opposite quantization contours.

A simple and effective technique to variationally interpret the structure of SUSY partners of mirror image potentials

Precise supersymmetric partner potentials can be generated for exactly solvable problems of the stationary Schrodinger equation. Construction of isospectral potential is not always possible for exactly solvable systems. This is a restriction, as most problems are not exactly solvable. Employment of mirror-image property can help to construct an exact isospectral partner of that potential. These potentials have chemical relevance to enantiomers. In this paper, we present a formulation as modelling to explore the form of SUSY pair of these potentials. Through polynomial fit, we correlate all possible basic SUSY partners and optimise it to best fit polynomial to present a typical energy value of N = 50.

Exactly solvable schrödinger equation for a class of multiparameter exponential‐type potentials

AbstractThe solution to a spectral problem involving the Schrödinger equation for a particular class of multiparameter exponential‐type potentials is presented. The proposal is based on the canonical transformation method applied to a general second‐order differential equation, multiplied by a functiong(x), to convert it into a Schrödinger‐like equation. The treatment of multiparameter exponential‐type potentials comes from the application of the transformed results to the hypergeometric equation under the assumption of a specificg(x). Besides presenting the explicit solutions and their spectral values, it is shown that the problem considered in this article unifies and generalizes several former studies. That is, the proposed exactly solvable multiparameter exponential‐type potential can be straightforwardly applied to particular exponential potentials depending on the choice of the involved parameters as exemplified for the Hulthén potential and their isospectral partner. Moreover, depending on the functiong(x), the proposal can be extended to find different exactly solvable potentials as well as to generate new potentials that could be useful in quantum chemical calculations. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012

A class of exactly solvable models to illustrate supersymmetry and test approximation methods in quantum mechanics

We obtain analytical expressions for the eigenvalues and eigenstates of a family of exponential-type potential wells. The supersymmetry formalism applied to quantum mechanics is summarized and illustrated by producing from this family of potentials another class of exact solutions made of their isospectral partners. A subset of the supersymmetric partners provides a class of exactly solvable double well potentials. The exact solutions of these potentials are used to test the robustness and accuracy of different approximation methods. We determine the ground state through a variational method applied to a set of trial functions and the entire spectrum using the WKB, JWKB, and its supersymmetric extension formulas. We comment on the importance of the Maslov index and on the range of validity of these semiclassical quantization approaches.

Supersymmetry in option pricing

We use supersymmetry to find the isospectral partners of Black–Scholes Hamiltonian without a potential and with a double knock out barrier potential. The pricing kernels for these Hamiltonians have also been obtained.

Isospectral potentials from modified factorization

Factorization of quantum-mechanical potentials has a long history extending back to the earliest days of the subject. In the present article, the nonuniqueness of the factorization is exploited to derive new isospectral nonsingular potentials. Many one-parameter families of potentials can be generated from known potentials using a factorization that involves superpotentials defined in terms of excited states of a potential. For these cases an operator representation is available. If ladder operators are known for the original potential, then a straightforward procedure exists for defining such operators for its isospectral partners. The generality of the method is illustrated with a number of examples which may have many possible applications in atomic and molecular physics.

Supersymmetry, PT-symmetry and spectral bifurcation

We demonstrate that large class of PT-symmetric complex potentials, which can have isospectral real partner potentials, possess two different superpotentials. In the parameter domain, where the superpotential is unique, the spectrum is real and shape-invariant, leading to translational shift in a suitable parameter by real units. The case of two different superpotentials, leading to same potential, yields broken PT-symmetry, the energy spectra in the two phases being separated by a bifurcation. Interestingly, these two superpotentials generate the two disjoint sectors of the Hilbert space. In the broken case, shape invariance produces complex parametric shifts.

Position dependent mass Schrödinger equation and isospectral potentials: Intertwining operator approach

Here, we have studied first- and second-order intertwining approaches to generate isospectral partner potentials of position dependent (effective) mass Schrödinger equation. The second-order intertwiner is constructed directly by taking it as second-order linear differential operator with position dependent coefficients, and the system of equations arising from the intertwining relationship is solved for the coefficients by taking an ansatz. A complete scheme for obtaining general solution is obtained, which is valid for any arbitrary potential and mass function. The proposed technique allows us to generate isospectral potentials with the following spectral modifications: (i) to add new bound state(s), (ii) to remove bound state(s), and (iii) to leave the spectrum unaffected. To explain our findings with the help of an illustration, we have used point canonical transformation to obtain the general solution of the position dependent mass Schrodinger equation corresponding to a potential and mass function. It is shown that our results are consistent with the formulation of type A N-fold supersymmetry [T. Tanaka, J. Phys. A 39, 219 (2006); A. Gonzalez-Lopez and T. Tanaka, J. Phys. A 39, 3715 (2006)] for the particular cases N=1 and N=2, respectively.

Isospectral potentials for the Schrödinger equation with a position‐dependent mass: Free‐particle potential model

AbstractIn this work, a method to obtain exactly solvable potentials for the one‐dimensional Schrödinger equation with a position‐dependent mass (SEPDM) and a procedure to determine their isospectral potential partners are presented. In the first case, the problem of exactly solvable SEPDM is worked by means of the point canonical transformation method which permit to convert the SEPDM problem into a standard Schrödinger‐like equation with a position‐independent mass (SLEPIM). In the second case, the procedure to obtain the partner isospectral potentials that fulfill with the SEPDM involves the Darboux transform applied to the SLEPIM. As example of the usefulness of the proposals, it is considered the special case of the free particle potential model as former potential of the SEPDM which leads to different exactly solvable potentials, and to their isospectral partners, depending on the choice of the position‐dependent mass distribution. The proposals are general and can be used in the search of those potentials leading to exactly solvable SEPDM and their isospectral partners, which could be useful in the modeling of quantum chemical properties in condensed matter applications. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Pseudo supersymmetric partners for the generalized Swanson model

New non Hermitian Hamiltonians are generated, as isospectral partners of the generalized Swanson model, viz., , where α, β are real constants, with α ≠ β, and and are generalized creation and annihilation operators. It is shown that the initial Hamiltonian H−, and its partner H+, are related by pseudo supersymmetry, and they share all the eigen energies except for the ground state. This pseudo supersymmetric extension enlarges the class of non Hermitian Hamiltonians H±, related to their respective Hermitian counterparts h±, through the same similarity transformation operator ρ:H± = ρ−1h±ρ. The formalism is applied to the entire class of shape-invariant models.

Fractional supersymmetry and hierarchy of shape invariant potentials

Fractional supersymmetric quantum mechanics is developed from a generalized Weyl-Heisenberg algebra. The Hamiltonian and the supercharges of fractional supersymmetric dynamical systems are built in terms of the generators of this algebra. The Hamiltonian gives rise to a hierarchy of isospectral Hamiltonians. Special cases of the algebra lead to dynamical systems for which the isospectral supersymmetric partner Hamiltonians are connected by a (translational or cyclic) shape invariance condition.

Supersymmetric quantum mechanics and statistical physics: Schrödinger-like thermodynamic equation

Supersymmetric quantum mechanics (SUSY QM) is used to determine the class of potentials that are solvable as well as to find their isospectral partners. To do that SUSY QM assumes the knowledge of the Witten superpotential as a particular solution of the Riccati equation involved in order to obtain the corresponding generalizations. Consequently, due to the fact that in statistical mechanics the internal energy is indeed known as a single temperature derivative of the Helmholtz free energy, a simple transformation has been proposed here in order to get, in an heuristic way, a Schrödinger-like thermodynamic (SLT) equation. In this proposal, the quantum Witten superpotential and the wavefunction correspond, in statistical physics, to the internal energy and to the canonical partition function respectively. With that, it is shown how all the results from SUSY QM can be incorporated in the statistical case, to find the generalized internal energy and partition functions as well as to generalize all the other related thermodynamical properties. As an example of the usefulness of the proposed SLT equation, the case of Boltzmann statistics has been considered explicitly and it was found that the corresponding generalized internal energy contain, as a particular case, the Bose–Einstein or the Fermi–Dirac probability distributions.

On the generation of non-Hermitian Hamiltonians with real spectra by a modified Darboux transform

Non- PT -symmetric, as well as PT -symmetric, non-Hermitian Hamiltonians with real spectra are generated by a modified Darboux transform. Isospectral partners of the harmonic oscillator are discussed as an example.

Exactly solvable combined potentials and their isospectral partners from SUSY

AbstractIn some quantum chemical applications, the potential models are linear combination of single exactly solvable potentials. This is the case, for example, of the equivalent of the Stark effect for a charged harmonic oscillator (HO) in a constant electric field or an HO in an external dipole field. The same occurs with some molecular systems where the determination of the vibrational spectra remains essentially as a numerical task; the Morse and Hulthén potentials are convenient models for the potential energy of a diatomic molecule. In this work, the supersymmetry (SUSY) and shape invariance methods used to determine exactly solvable potentials are extended to obtain, in a closed form, the eigenvalues of exactly solvable combined potentials, as well as their isospectral or generalized partner potentials. Essentially, the proposal is based on the existence of the Witten superpotential [W(x)] associated with single exactly solvable potentials to find the Witten superpotential for the combined potential under study. For instance, the assumption of two W(x) allows to obtain an exactly solvable combined potential which is given by a linear combination between the former potentials plus a term resulting from the product of the two single W(x). The corresponding generalized partner potentials comes from the involved Riccati equation when the resulting Witten superpotential for the combined potential is taken as a specific solution. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

SECOND-ORDER INTERTWINING OPERATOR AND ISOSPECTRAL PARTNER OF COMPLEX PT INVARIANT POTENTIAL

A generalized second-order differential realization of intertwining operator is given and isospectral partners of complex PT invariant Scarf II potential are obtained.

Isospectral orthogonal polynomials from the Darboux transforms

AbstractOrthogonal polynomials (OP) are used in many branches of the mathematical and physical sciences; in particular they are part of the eigenfunctions of quantum chemical (QC) potential models. Recently, in the search for new solvable potentials to be useful in QC applications, the use of supersymmetry (SUSY) and Hamiltonian intertwining methods have shown the existence of isospectral potential partners. Also, it has been proved that SUSY is equivalent to the standard Darboux transform (DT) applied to Sturm–Liouville‐type problems. Consequently, because of the similarity between the differential equations of OP and of SL, we consider the application of the standard and generalized DT to find the isospectral OP that are partners of the standard one. To attain this purpose, we use a pair of transformations, a point canonical and a gauge, to convert the SL differential equation into a Schrödinger‐like equation allowing OP solutions. As a useful application of our proposal, we consider some of the most important OP by obtaining the generalized isospectral OP partners. Also, we shown the associated isospectral QC potentials for each of the orthogonal polynomials considered in this work. The proposed procedure can be used to obtain new orthogonal polynomials as well as to find new solvable QC potential models. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004