skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Quantum models with energy-dependent potentials solvable in terms of exceptional orthogonal polynomials

Abstract

We construct energy-dependent potentials for which the Schrödinger equations admit solutions in terms of exceptional orthogonal polynomials. Our method of construction is based on certain point transformations, applied to the equations of exceptional Hermite, Jacobi and Laguerre polynomials. We present several examples of boundary-value problems with energy-dependent potentials that admit a discrete spectrum and the corresponding normalizable solutions in closed form.

Authors:
 [1];  [2];  [3]
  1. Department of Mathematics and Actuarial Science, Indiana University Northwest, 3400 Broadway, Gary IN 46408 (United States)
  2. (United States)
  3. Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108 (India)
Publication Date:
OSTI Identifier:
22617483
Resource Type:
Journal Article
Resource Relation:
Journal Name: Annals of Physics; Journal Volume: 378; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOUNDARY-VALUE PROBLEMS; LAGUERRE POLYNOMIALS; MATHEMATICAL SOLUTIONS; SCHROEDINGER EQUATION

Citation Formats

Schulze-Halberg, Axel, E-mail: axgeschu@iun.edu, Department of Physics, Indiana University Northwest, 3400 Broadway, Gary IN 46408, and Roy, Pinaki, E-mail: pinaki@isical.ac.in. Quantum models with energy-dependent potentials solvable in terms of exceptional orthogonal polynomials. United States: N. p., 2017. Web. doi:10.1016/J.AOP.2017.01.023.
Schulze-Halberg, Axel, E-mail: axgeschu@iun.edu, Department of Physics, Indiana University Northwest, 3400 Broadway, Gary IN 46408, & Roy, Pinaki, E-mail: pinaki@isical.ac.in. Quantum models with energy-dependent potentials solvable in terms of exceptional orthogonal polynomials. United States. doi:10.1016/J.AOP.2017.01.023.
Schulze-Halberg, Axel, E-mail: axgeschu@iun.edu, Department of Physics, Indiana University Northwest, 3400 Broadway, Gary IN 46408, and Roy, Pinaki, E-mail: pinaki@isical.ac.in. Wed . "Quantum models with energy-dependent potentials solvable in terms of exceptional orthogonal polynomials". United States. doi:10.1016/J.AOP.2017.01.023.
@article{osti_22617483,
title = {Quantum models with energy-dependent potentials solvable in terms of exceptional orthogonal polynomials},
author = {Schulze-Halberg, Axel, E-mail: axgeschu@iun.edu and Department of Physics, Indiana University Northwest, 3400 Broadway, Gary IN 46408 and Roy, Pinaki, E-mail: pinaki@isical.ac.in},
abstractNote = {We construct energy-dependent potentials for which the Schrödinger equations admit solutions in terms of exceptional orthogonal polynomials. Our method of construction is based on certain point transformations, applied to the equations of exceptional Hermite, Jacobi and Laguerre polynomials. We present several examples of boundary-value problems with energy-dependent potentials that admit a discrete spectrum and the corresponding normalizable solutions in closed form.},
doi = {10.1016/J.AOP.2017.01.023},
journal = {Annals of Physics},
number = ,
volume = 378,
place = {United States},
year = {Wed Mar 15 00:00:00 EDT 2017},
month = {Wed Mar 15 00:00:00 EDT 2017}
}
  • We introduce a method for constructing Darboux (or supersymmetric) pairs of pseudoscalar and scalar Dirac potentials that are associated with exceptional orthogonal polynomials. Properties of the transformed potentials and regularity conditions are discussed. As an application, we consider a pseudoscalar Dirac potential related to the Schrödinger model for the rationally extended radial oscillator. The pseudoscalar partner potentials are constructed under the first- and second-order Darboux transformations.
  • Quantum systems consisting of a linear chain of n harmonic oscillators coupled by a quadratic nearest-neighbour interaction are considered. We investigate when such a system is analytically solvable, in the sense that the eigenvalues and eigenvectors of the interaction matrix have analytically closed expressions. This leads to a relation with Jacobi matrices of systems of discrete orthogonal polynomials. Our study is first performed in the case of canonical quantization. Then we consider these systems under Wigner quantization, leading to solutions in terms of representations of Lie superalgebras. Finally, we show how such analytically solvable Hamiltonians also play a role inmore » another application, that of spin chains used as communication channels in quantum computing. In this context, the analytic solvability leads to closed form expressions for certain transition amplitudes.« less
  • Within the context of supersymmetric quantum mechanics and its related hierarchies of integrable quantum Hamiltonians and potentials, a general programme is outlined and applied to its first two simplest illustrations. Going beyond the usual restriction of shape invariance for intertwined potentials, it is suggested to require a similar relation for Hamiltonians in the hierarchy separated by an arbitrary number of levels, N. By requiring further that these two Hamiltonians be in fact identical up to an overall shift in energy, a periodic structure is installed in the hierarchy which should allow for its resolution. Specific classes of orthogonal polynomials characteristicmore » of such periodic hierarchies are thereby generated, while the methods of supersymmetric quantum mechanics then lead to generalised Rodrigues formulae and recursion relations for such polynomials. The approach also offers the practical prospect of quantum modelling through the engineering of quantum potentials from experimental energy spectra. In this paper, these ideas are presented and solved explicitly for the cases N= 1 and N= 2. The latter case is related to the generalised Laguerre polynomials, for which indeed new results are thereby obtained. In the context of dressing chains and deformed polynomial Heisenberg algebras, some partial results for N⩾ 3 also exist in the literature, which should be relevant to a complete study of the N⩾ 3 general periodic hierarchies.« less
  • By using the properties of orthogonal polynomials, we present an exact unitary transformation that maps the Hamiltonian of a quantum system coupled linearly to a continuum of bosonic or fermionic modes to a Hamiltonian that describes a one-dimensional chain with only nearest-neighbor interactions. This analytical transformation predicts a simple set of relations between the parameters of the chain and the recurrence coefficients of the orthogonal polynomials used in the transformation and allows the chain parameters to be computed using numerically stable algorithms that have been developed to compute recurrence coefficients. We then prove some general properties of this chain systemmore » for a wide range of spectral functions and give examples drawn from physical systems where exact analytic expressions for the chain properties can be obtained. Crucially, the short-range interactions of the effective chain system permit these open-quantum systems to be efficiently simulated by the density matrix renormalization group methods.« less
  • In recent years, many exceptional orthogonal polynomials (EOP) were introduced and used to construct new families of 1D exactly solvable quantum potentials, some of which are shape invariant. In this paper, we construct from Hermite and Laguerre EOP and their related quantum systems new 2D superintegrable Hamiltonians with higher-order integrals of motion and the polynomial algebras generated by their integrals of motion. We obtain the finite-dimensional unitary representations of the polynomial algebras and the corresponding energy spectrum. We also point out a new type of degeneracies of the energy levels of these systems that is associated with holes in sequencesmore » of EOP.« less