Sharp spectral stability for a class of singularly perturbed pseudo-differential operators

Abstract

Let $a(x,\xi )$ be a real Hörmander symbol of the type $S_{0,0}^0({\mathbb{R}}^d\times {\mathbb{R}}^d)$, let $F$ be a smooth function with all its derivatives globally bounded, and let $K_{\delta }$ be the self-adjoint Weyl quantization of the perturbed symbols $a(x+F(\delta x),\xi )$, where $|\delta |\le 1$. First, we prove that the Hausdorff distance between the spectra of $K_{\delta }$ and $K_0$ is bounded by $\sqrt{|\delta |}$, and we give examples where spectral gaps of this magnitude can open when $\delta \ne 0$. Second, we show that the distance between the spectral edges of $K_{\delta }$ and $K_0$ (and also the edges of the inner spectral gaps, as long as they remain open at $\delta =0$) are of order $|\delta |$, and give a precise dependence on the width of the spectral gaps.