Exactly solvable nonseparable and nondiagonalizable two-dimensional model with quadratic complex interaction

Abstract

We study a quantum model with nonisotropic two-dimensional oscillator potential but with additional quadratic interaction x1x2x1x2 with imaginary coupling constant. It is shown that for a specific connection between coupling constant and oscillator frequencies, the modelis not amenable to a conventional separation of variables. The property of shape invariance allows to find analytically all eigenfunctions and the spectrum is found to be equidistant. It is shown that the Hamiltonian is nondiagonalizable, and the resolution of the identity must include also the corresponding associated functions. These functions are constructed explicitly, and their properties are investigated. The problem of RR-separation of variables in two-dimensional systems is discussed.