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