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

TL;DR

This paper investigates spectral stability of certain pseudodifferential operators under perturbations, removing previous assumptions and establishing how spectral edges vary with perturbation size based on decay rates of derivatives.

Contribution

It extends prior work by removing the 'slow variation' assumption, analyzing spectral edge behavior for a broader class of symbols with bounded derivatives.

Findings

Spectral edge distances behave like |elta|^6 for small elta.

Hausdorff distance between spectra remains of order 6.

Decay rate of second derivatives influences spectral edge behavior.

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 $a\big(x+\delta\,F(x),\xi\big)$ with $a$ a real H\"{o}rmander 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 $|\delta|\leq1$. 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 $\sqrt{|\delta|}$, the distance between their spectral edges behaves like $|\delta|^\nu$ with $\nu\in[1/2,1)$ depending on the rate of decay of the second derivatives of $F$ at infinity.