Modification of Crum's Theorem for `Discrete' Quantum Mechanics

Abstract

Crum's theorem in one-dimensional quantum mechanics asserts the existence of an associated Hamiltonian system for any given Hamiltonian with the complete set of eigenvalues and eigenfunctions. The associated system is iso-spectral to the original one except for the lowest energy state, which is deleted. A modification due to Krein-Adler provides algebraic construction of a new complete Hamiltonian system by deleting a finite number of energy levels. Here we present a discrete version of the modification based on the Crum's theorem for the `discrete' quantum mechanics developed by two of the present authors.

31 pages, 2 figures. Two typos (eq.(3.16), ref.[15]) corrected, several comments added. To appear in Progress of Theoretical Physics

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Cited by in corpus (32)

• Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials
• Krein-Adler transformations for shape-invariant potentials and pseudo virtual states
• Discrete Quantum Mechanics
• 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
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials
• Dual Christoffel transformations
• Prepotential approach to solvable rational extensions of Harmonic Oscillator and Morse potentials
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : III
• Casoratian Identities for the Wilson and Askey-Wilson Polynomials
• Scattering Amplitudes for Multi-indexed Extensions of Solvable Potentials
• Equivalences of the Multi-Indexed Orthogonal Polynomials
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : II
• Exceptional orthogonal polynomials and new exactly solvable potentials in quantum mechanics
• Global Solutions of Certain Second-Order Differential Equations with a High Degree of Apparent Singularity
• 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
• Exactly solvable potentials with finitely many discrete eigenvalues of arbitrary choice
• Solvable Discrete Quantum Mechanics: q-Orthogonal Polynomials with |q|=1 and Quantum Dilogarithm
• Exactly and quasi-exactly solvable `discrete' quantum mechanics
• New Determinant Expressions of the Multi-indexed Orthogonal Polynomials in Discrete Quantum Mechanics
• Dual Polynomials of the Multi-Indexed ($q$-)Racah Orthogonal Polynomials
• Non-polynomial extensions of solvable potentials a la Abraham-Moses
• Wronskian/Casoratian Identities and their Application to Quantum Mechanical Systems
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials V : Racah and $q$-Racah types
• Reflectionless Potentials for Difference Schrödinger Equations
• Casoratian Identities for the Discrete Orthogonal Polynomials in Discrete Quantum Mechanics with Real Shifts
• Exactly Solvable Quantum Mechanics
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials VI : Meixner-Pollaczek and continuous Hahn types
• Discrete Orthogonality Relations for the Multi-Indexed Orthogonal Polynomials in Discrete Quantum Mechanics with Pure Imaginary Shifts