Crum's Theorem for `Discrete' Quantum Mechanics

Abstract

In one-dimensional quantum mechanics, or the Sturm-Liouville theory, Crum's theorem describes the relationship between the original and the associated Hamiltonian systems, which are iso-spectral except for the lowest energy state. Its counterpart in `discrete' quantum mechanics is formulated algebraically, elucidating the basic structure of the discrete quantum mechanics, whose Schrödinger equation is a difference equation.

13 pages, to be published in Prog.Theor.Phys., several comments and references added

References in corpus (1)

• Exactly solvable `discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states

Cited by in corpus (30)

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• Discrete Quantum Mechanics
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• Multi-indexed (q-)Racah Polynomials
• Extensions of solvable potentials with finitely many discrete eigenstates
• The Exceptional (X_{\ell}) (q)-Racah Polynomials
• Exceptional Askey-Wilson type polynomials through Darboux-Crum transformations
• Orthogonal Polynomials from Hermitian Matrices II
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials
• Dual Christoffel transformations
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : III
• Casoratian Identities for the Wilson and Askey-Wilson Polynomials
• Unified theory of exactly and quasi-exactly solvable `Discrete' quantum mechanics: I. Formalism
• Equivalences of the Multi-Indexed Orthogonal Polynomials
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : II
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials IV : closure relations and creation/annihilation operators
• Exactly Solvable Discrete Quantum Mechanical Systems and Multi-indexed Orthogonal Polynomials of the Continuous Hahn and Meixner-Pollaczek Types
• Solvable Discrete Quantum Mechanics: q-Orthogonal Polynomials with |q|=1 and Quantum Dilogarithm
• New Determinant Expressions of the Multi-indexed Orthogonal Polynomials in Discrete Quantum Mechanics
• Exactly and quasi-exactly solvable `discrete' quantum mechanics
• Dual Polynomials of the Multi-Indexed ($q$-)Racah Orthogonal Polynomials
• Reflectionless Potentials for Difference Schrödinger Equations
• Non-polynomial extensions of solvable potentials a la Abraham-Moses
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials V : Racah and $q$-Racah types
• Hypercomplex Fock States for Discrete Electromagnetic Schrödinger Operators: A Bayesian Probability Perspective
• Casoratian Identities for the Discrete Orthogonal Polynomials in Discrete Quantum Mechanics with Real Shifts
• Discrete Orthogonality Relations for the Multi-Indexed Orthogonal Polynomials in Discrete Quantum Mechanics with Pure Imaginary Shifts
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials VI : Meixner-Pollaczek and continuous Hahn types
• Some Difference Relations for Orthogonal Polynomials of a Continuous Variable in the Askey Scheme