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.

Two families of superintegrable and isospectral potentials in two dimensions

“…We end this section by explaining our use of the adjective "isospectral". Two Hamiltonians are said to be isospectral if they have the same eigenvalue spectrum [18,22,23]. In this sense two linearly intertwined Hamiltonians are always formally isospectral except the eigenvalues corresponding to the kernel of the intertwining operator.…”

“…where we have made use of ∇ 2 L 0 = g ′′ (u) and of the second equation of (23). Note that we could have chosen φ = 0, but since it costs almost nothing we keep φ in our formulae in order to see that action of E(2).…”

“…As a result we have found a five parameter (a, a 1 , λ 1 , λ 2 , φ) family of 2D potentials that are generated, in a nontrivial way, by two 1D problems. We specify a second subfamily of potentials by taking, in (23)(24) and (27) …”

Localization of a nonlocal interaction

“…The phase shifts are to be described by what is called a spectral function and the potential is derived from the kernel that solves their integral equation. Later on, the algorithms of the inverse scattering problem, within the framework of supersymmetric quantum mechanics, have been exploited by a number of researchers [6][7][8][9][10] to generate phase equivalent potentials.…”

A Novel Isospectral Deformation Chain in Supersymmetric Quantum Mechanics

“…The dominating approach is to study isospectral Hamilton operators. Different operator methods exist, but the main ones was brought under a single unifying principle by Pursey [4] with the use of isometric operators. A second approach to the study of isospectral transformations is what has been called deformation theory (see [5], e.g.).…”