Quasi-exactly solvable cases of the N-dimensional symmetric quartic anharmonic oscillator

TL;DR

This paper identifies specific conditions under which the N-dimensional symmetric quartic anharmonic oscillator becomes exactly solvable, providing explicit solutions and analytical expressions for certain excited states.

Contribution

It introduces a set of special parameter conditions that render the N-dimensional symmetric quartic anharmonic oscillator quasi-exactly solvable, with explicit matrix solutions and polynomial state descriptions.

Findings

Exact solutions exist under specific parameter conditions.

Explicit finite-dimensional matrix equations are constructed.

Analytical expressions for some excited states are provided.

Abstract

The O(N) invariant quartic anharmonic oscillator is shown to be exactly solvable if the interaction parameter satisfies special conditions. The problem is directly related to that of a quantum double well anharmonic oscillator in an external field. A finite dimensional matrix equation for the problem is constructed explicitly, along with analytical expressions for some excited states in the system. The corresponding Niven equations for determining the polynomial solutions for the problem are given.