Modification of Crums Theorem for `Discrete Quantum Mechanics

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.

💡 Research Summary

The paper extends the well‑known Crum theorem and its Krein‑Adler generalization from ordinary one‑dimensional quantum mechanics to the framework of discrete quantum mechanics (DQM). In the continuous setting, Crum’s theorem provides a systematic way to construct a new Hamiltonian H