Spectral regularity with respect to dilations for a class of pseudodifferential operators

We continue the study of the perturbation problem discussed in \cite{CP3} and get rid of the ‘slow variation’ assumption by considering symbols of the form $a(x+\delta F(x),\xi )$ with $a$ a real Hörmander symbol of class $S_{0,0}^0({\mathbb{R}}^d\times {\mathbb{R}}^d)$ and $F$ a smooth function with all its derivatives globally bounded, with $\mathrm{\mid }\delta \mathrm{\mid }\le 1$. We prove that while the Hausdorff distance between the spectra of the Weyl quantization of the above symbols in a neighbourhood of $\delta =0$ is still of the order $\sqrt{\mathrm{\mid }\delta \mathrm{\mid }}$, the distance between their spectral edges behaves like $\mathrm{\mid }\delta {\mathrm{\mid }}^{\nu }$ with $\nu \in [1\mathrm{/}2,1)$ depending on the rate of decay of the second derivatives of $F$ at infinity.