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

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

2010-04-02

arxiv.org/abs/1004.0289v2

Prog. Theor. Phys. 124 (2010), 1-26

Physics

Mathematical Physics

31 pages, 2 figures. Two typos (eq.(3.16), ref.[15]) corrected, several comments added. To appear in Progress of Theoretical P

Scientific paper

10.1143/PTP.124.1

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.

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