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arXiv · hep-th/0201080

New Methods for Two-Dimensional Schrödinger Equation: SUSY-separation of Variables and Shape Invariance

F. Cannata·M. V. Ioffe·D. N. Nishnianidze
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Abstract

Two new methods for investigation of two-dimensional quantum systems, whose Hamiltonians are not amenable to separation of variables, are proposed. 1)The first one - $SUSY-$ separation of variables - is based on the intertwining relations of Higher order SUSY Quantum Mechanics (HSUSY QM) with supercharges allowing for separation of variables. 2)The second one is a generalization of shape invariance. While in one dimension shape invariance allows to solve algebraically a class of (exactly solvable) quantum problems, its generalization to higher dimensions has not been yet explored. 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 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.

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F. Cannata, M. V. Ioffe, D. N. Nishnianidze. 2002-01-13. New Methods for Two-Dimensional Schrödinger Equation: SUSY-separation of Variables and Shape Invariance. https://doi.org/10.1088/0305-4470%2F35%2F6%2F305

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