The Pseudospectrum of Systems of Semiclassical Operators

TL;DR

This paper investigates the pseudospectra of non-selfadjoint semiclassical differential operator systems, extending scalar operator results, and establishes resolvent blow-up behavior and estimates for quasi-symmetrizable and subelliptic systems.

Contribution

It generalizes pseudospectrum analysis from scalar to systems of semiclassical operators, introducing new classes and proving resolvent estimates.

Findings

Resolvent blows up outside a nowhere dense set of degenerate values.

Defined quasi-symmetrizable and subelliptic systems with resolvent estimates.

Extended scalar pseudospectrum results to operator systems.

Abstract

The pseudospectra (or spectral instability) of non-selfadjoint operators is a topic of current interest in applied mathematics. In fact, for non-selfadjoint operators the resolvent could be very large outside the spectrum, making the numerical computation of the complex eigenvalues very hard. This has importance, for example, in quantum mechanics, random matrix theory and fluid dynamics. The occurence of pseudospectra for non-selfadjoint semiclassical differential operators is due to the existence of quasimodes, i.e., approximate local solutions to the eigenvalue problem. For scalar operators, the quasimodes appear since the bracket condition is not satisfied for topological reasons, see the paper by Dencker, Sjostrand and Zworski in Comm. Pure Appl. Math. 57 (2004), 384-415. In this paper we shall investigate how these result can be generalized to square systems of semiclassical differential operators of principal type. These are the systems whose principal symbol vanishes of first order on its kernel. We show that the resolvent blows up as in the scalar case, except in a nowhere dense set of degenerate values. We also define quasi-symmetrizable systems and systems of subelliptic type for which we prove estimates on the resolvent.