Crums Theorem for `Discrete Quantum Mechanics

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.

💡 Research Summary

The paper presents a rigorous algebraic formulation of Crum’s theorem within the framework of discrete quantum mechanics (DQM), where the Schrödinger equation is replaced by a finite‑difference equation on a lattice. After reviewing the classical continuous‑variable version—where a Hamiltonian H can be factorized as H = A†A and a partner Hamiltonian H