On the pseudospectrum of elliptic quadratic differential operators

TL;DR

This paper investigates the pseudospectrum of elliptic quadratic differential operators, identifying conditions for spectral stability and describing regions of spectral instability with geometric precision, supported by numerical simulations.

Contribution

It provides a necessary and sufficient condition for spectral stability of these operators and characterizes the regions of spectral instability at high energies.

Findings

Spectral stability is guaranteed under a specific condition on the Weyl symbol.

Violating this condition leads to strong spectral instabilities at high energies.

Numerical simulations reveal false eigenvalues far from the actual spectra.

Abstract

We study the pseudospectrum of a class of non-selfadjoint differential operators. Our work consists in a detailed study of the microlocal properties, which rule the spectral stability or instability phenomena appearing under small perturbations for elliptic quadratic differential operators. The class of elliptic quadratic differential operators stands for the class of operators defined in the Weyl quantization by complex-valued elliptic quadratic symbols. We establish in this paper a simple necessary and sufficient condition on the Weyl symbol of these operators, which ensures the stability of their spectra. When this condition is violated, we prove that it occurs some strong spectral instabilities for the high energies of these operators, in some regions which can be far away from their spectra. We give a precise geometrical description of them, which explains the results obtained for these operators in some numerical simulations giving the computation of false eigenvalues far from their spectra by algorithms for eigenvalues computing.