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

TL;DR

This paper extends Crum's theorem to discrete quantum mechanics, enabling algebraic construction of new Hamiltonian systems by deleting specific energy levels, thus broadening the theoretical framework for discrete quantum systems.

Contribution

It introduces a discrete version of Crum's theorem, adapting the Krein-Adler modification for discrete quantum mechanics, which was not previously established.

Findings

Developed a discrete Crum's theorem for quantum mechanics

Constructed new Hamiltonians by deleting energy levels

Extended algebraic methods to discrete quantum systems

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.