Isometric operators, isospectral Hamiltonians, and supersymmetric quantum mechanics

Abstract

Isometric operators are used to provide a unified theory of the three established procedures for generating one-parameter families of isospectral Hamiltonians. All members of the same family of isospectral Hamiltonians are unitarily equivalent, and the unitary transformations between them form a group isomorphic with the additive group of real numbers. The theory is generalized by including the parameter identifying a member of an isospectral family as a new variable. The unitary transformations within a family correspond to translations in the parameter space. The generator of infinitesimal translations represents a conserved quantity in the extended theory. Isometric operators are then applied to the development of models of supersymmetric quantum mechanics. In addition to the standard models based on the Darboux procedure, I show how to construct models based on the Abraham-Moses and Pursey procedures. The formalism shows that the Nieto ambiguity present in all models of supersymmetric quantum mechanics can be interpreted as a renormalization of the ground state of the supersymmetric system. This allows a generalization of supersymmetric quantum mechanics analogous to that developed for systems of isospectral Hamiltonians.