Equidistance of the Complex 2-Dim Anharmonic Oscillator Spectrum: Exact Solution

TL;DR

This paper analyzes a class of complex 2D quantum models with shape invariance, revealing equidistant spectra, non-diagonalizability, and providing explicit solutions for anharmonic interactions.

Contribution

It introduces a new class of exactly solvable complex 2D quantum models with shape invariance and constructs explicit eigenfunctions and associated functions.

Findings

Spectrum is equidistant due to shape invariance.

Hamiltonian is non-diagonalizable, requiring associated functions.

Explicit solutions are provided for anharmonic second-plus-fourth order interactions.

Abstract

We study a class of quantum two-dimensional models with complex potentials of specific form. They can be considered as the generalization of a recently studied model with quadratic interaction not amenable to conventional separation of variables. In the present case, the property of shape invariance provides the equidistant form of the spectrum and the algorithm to construct eigenfunctions analytically. It is shown that the Hamiltonian is non-diagonalizable, and the resolution of identity must include also the corresponding associated functions. In the specific case of anharmonic second-plus-fourth order interaction, expressions for the wave functions and associated functions are constructed explicitly for the lowest levels, and the recursive algorithm to produce higher level wave functions is given.