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

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.