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

Abstract

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 $a\big(x+δ\,F(x),ξ\big)$ 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 $|δ|\leq1$. We prove that while the Hausdorff distance between the spectra of the Weyl quantization of the above symbols in a neighbourhood of $δ=0$ is still of the order $\sqrt{|δ|}$, the distance between their spectral edges behaves like $|δ|^ν$ with $ν\in[1/2,1)$ depending on the rate of decay of the second derivatives of $F$ at infinity.