The quantum harmonic oscillator with icosahedral symmetry and some explicit wavefunctions

TL;DR

This paper explores a quantum harmonic oscillator with icosahedral symmetry using Dunkl Laplacian, defining special wavefunctions, calculating norms, and analyzing symmetry-invariant solutions with explicit formulas.

Contribution

It introduces explicit wavefunctions and norm calculations for an icosahedral symmetric quantum oscillator using Dunkl operators, extending known models to new symmetry groups.

Findings

Explicit wavefunctions derived from icosahedral vertices

Squared norms computed for polynomial wavefunctions

Identification of commuting sixth-order operator

Abstract

The Dunkl Laplacian is used to define the Hamiltonian of a modified quantum harmonic oscillator, associated with any finite reflection group. The potential is a sum of the inverse squares of the linear functions whose zero sets are the mirrors of the group's reflections. The symmetric group version of this is known as the Calogero-Moser model of N identical particles on a line. This paper focuses on the group of symmetries of the regular icosahedron, associated to the root system of type H3. Special wavefunctions are defined by a generating function arising from the vertices of the icosahedron and have the key property of allowing easy calculation of the effect of the Dunkl Laplacian. The ground state is the product of a Gaussian function with powers of linear functions coming from the root system. Two types of wavefunctions are considered, inhomogeneous polynomials with specified top-degree part, and homogeneous harmonic polynomials. The squared norms for both types are explicitly calculated. Symmetrization is applied to produce the invariant polynomials of both types, as well as their squared norms. The action of the angular momentum square on the harmonic homogeneous polynomials is determined. There is also a sixth-order operator commuting with the Hamiltonian and the group action.