TY - JOUR

T1 - New methods for the two-dimensional Schrödinger equation

T2 - SUSY-separation of variables and shape invariance

AU - Cannata, F.

AU - Ioffe, M. V.

AU - Nishnianidze, D. N.

PY - 2002/2/15

Y1 - 2002/2/15

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0037085113&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/35/6/305

DO - 10.1088/0305-4470/35/6/305

M3 - Article

AN - SCOPUS:0037085113

VL - 35

SP - 1389

EP - 1404

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 6

ER -