Refined Weyl law for homogeneous perturbations of the harmonic oscillator

TL;DR

This paper refines the Weyl law for the harmonic oscillator with certain perturbations, showing weaker singularities at nonzero multiples of 2π and improving the remainder estimate in the spectral counting function.

Contribution

It introduces a class of perturbations where the spectral trace singularities are weaker, leading to an improved remainder estimate in the Weyl law for the harmonic oscillator.

Findings

Singularities at nonzero multiples of 2π are weaker under certain perturbations.

Remainder term in Weyl law is o(λ^{d-1}), better than previous O(λ^{d-1}).

Results hold for dimension d ≥ 2 with isotropic pseudodifferential perturbations.

Abstract

Let $H$ denote the harmonic oscillator Hamiltonian on $\mathbb{R}^d,$ perturbed by an isotropic pseudodifferential operator of order $1.$ We consider the Schr\"odinger propagator $U(t)=e^{-itH},$ and find that while $\operatorname{singsupp} \operatorname{Tr} U(t) \subset 2 \pi \mathbb{Z}$ as in the unperturbed case, there exists a large class of perturbations in dimension $d \geq 2$ for which the singularities of $\operatorname{Tr} U(t)$ at nonzero multiples of $2 \pi$ are weaker than the singularity at $t=0$. The remainder term in the Weyl law is of order $o(\lambda^{d-1})$, improving in these cases the $O(\lambda^{d-1})$ remainder previously established by Helffer--Robert.