Crum's Theorem for `Discrete' Quantum Mechanics

TL;DR

This paper extends Crum's theorem to discrete quantum mechanics, providing an algebraic formulation that clarifies the structure of difference-equation-based Schrödinger systems, highlighting their spectral relationships.

Contribution

It introduces an algebraic version of Crum's theorem for discrete quantum mechanics, revealing the spectral and structural properties of difference equation systems.

Findings

Established an algebraic framework for discrete Crum's theorem

Demonstrated the spectral relationship between original and associated systems

Clarified the structure of discrete Schrödinger equations

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 Schr\"odinger equation is a difference equation.