Self-adjoint realization of the harmonic oscillator in polar coordinates and some consequences

TL;DR

This paper explores the spectral decomposition of the harmonic oscillator in multiple dimensions, providing new proofs of its rotational symmetry and related identities using functional analysis.

Contribution

It offers new, simpler proofs of known properties of harmonic oscillator eigenfunctions and their symmetries through functional analysis techniques.

Findings

Spectral decomposition in different bases

Rotational symmetry of Hermite projections

Hecke-Bochner type identity for harmonic oscillator

Abstract

We consider spectral decomposition of the harmonic oscillator in $\mathbb R^n$ in terms of two different orthonormal bases in $L^2(\mathbb R^n)$ consisting of its eigenfunctions. Then, using purely functional analysis tools we provide simple proofs of rotational symmetry of the Hermite projection operators studied by Kochneff, and Thangavelu's Hecke-Bochner type identity.