Perturbation Theory for Quantum Mechanics in its Hessenberg-Matrix Representation

Abstract

For years, the partial integrability in quantum mechanics [e.g., the exceptional terminating Lanczos solutions or the so called quasi-exactly solvable "next-to-elementary" systems] represented a challenge in perturbation theory. The main difficulty lied in an incompleteness of the available zero-order wavefunctions and in the related impossibility of an easy construction of the necessary zero-order propagators. We describe a solution of this problem, based on an unusual choice of model space. A few examples illustrate the underlying technicalities: Paying detailed attention to the Hessenberg-matrix Hamiltonians, our formalism compensates the incompleteness of integrability by a slightly less elementary (viz., non-diagonal but still triangular) matrix structure of its propagators.