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

Let $a(x,\xi)$ be a real H\"ormander 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 $|\delta|\leq 1$. 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 $\delta\neq 0$. 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.