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

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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 Crum's theorem for the 'discrete' quantum mechanics developed by two of the present authors.

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PROGRESS OF THEORETICAL PHYSICS. 124(1):1-26 (2010)

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