Additional restrictions on quasi-exactly solvable systems

TL;DR

This paper investigates boundary constraints on 2D quantum systems, revealing that hermiticity limits their solvability to quasi-exactly solvable, but pseudo-Hermitian cases can be fully solvable.

Contribution

It introduces new families of 2D quantum systems and analyzes how hermiticity constraints affect their exact solvability, including pseudo-Hermitian cases.

Findings

Boundary constraints lead to quasi-exact solvability in real systems.

Some pseudo-Hermitian systems are exactly solvable.

New families of solvable 2D quantum systems are constructed.

Abstract

In this paper we discuss constraints on two-dimensional quantum-mechanical systems living in domains with boundaries. The constrains result from the requirement of hermicity of corresponding Hamiltonians. We construct new two-dimensional families of formally exactly solvable systems and applying such constraints show that in real the systems are quasi-exactly solvable at best. Nevertheless in the context of pseudo-Hermitian Hamiltonians some of the constructed families are exactly solvable.