Exactly Solvable Non-Separable and Non-Diagonalizable 2-Dim Model with Quadratic Complex Interaction

TL;DR

This paper introduces an exactly solvable two-dimensional quantum model with quadratic complex interaction that cannot be separated conventionally, revealing non-diagonalizability and requiring associated functions for a complete spectral description.

Contribution

It presents a novel non-separable, non-diagonalizable 2D quantum model with explicit eigenfunctions, spectrum, and associated functions, expanding understanding of complex interactions in quantum systems.

Findings

Eigenfunctions and spectrum are analytically obtained.

The Hamiltonian is non-diagonalizable, requiring associated functions.

The model exhibits shape invariance and equidistant spectrum.

Abstract

We study a quantum model with non-isotropic two-dimensional oscillator potential but with additional quadratic interaction $x_1x_2$ with imaginary coupling constant. It is shown, that for a specific connection between coupling constant and oscillator frequences, the model {\it is 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 non-diagonalizable, 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 $R-$separation of variables in two-dimensional systems is discussed.