2002/01/13 by F. Cannata, M. V. Ioffe, D. N. Nishnianidze · 7 citations
Mathematics · Physics and Astronomy · #Nonlinear Photonic Systems #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #hep-th #math-ph #math.MP #nlin.SI #quant-ph
paper · pdf · doi:10.1088/0305-4470/35/6/305
published as J.Phys.A35:1389-1404,2002 · 25 pages, Latex, Journal of Physics A, to be published
arxiv created 2002/01/13 · openalex publication_date 2002/02/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
Two new methods for the investigation of two-dimensional quantum systems, whose Hamiltonians are not amenable to separation of variables, are proposed. The first one—SUSY-separation of variables—is based on the intertwining relations of higher-order SUSY quantum mechanics (HSUSY QM) with supercharges allowing separation of variables. The second one is a generalization of shape invariance. While in one dimension shape invariance allows us to solve algebraically a class of (exactly solvable) quantum problems, its generalization to higher dimensions has not been explored yet. Here we provide a formal framework in HSUSY QM for two-dimensional quantum mechanical systems for which shape invariance holds. Given the knowledge of one eigenvalue and eigenfunction, shape invariance allows us to construct a chain of new eigenfunctions and eigenvalues. These methods are applied to a two-dimensional quantum system, and partial explicit solvability is achieved in the sense that only part of the spectrum is found analytically and a limited set of eigenfunctions is constructed explicitly.