Vol. 1, No. 3, 2008

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The pseudospectrum of systems of semiclassical operators

Nils Dencker

Vol. 1 (2008), No. 3, 323–373

The pseudospectrum (or spectral instability) of non-self-adjoint operators is a topic of current interest in applied mathematics. In fact, for non-self-adjoint operators the resolvent could be very large outside the spectrum, making numerical computation of the complex eigenvalues very hard. This has importance, for example, in quantum mechanics, random matrix theory and fluid dynamics.

The occurrence of false eigenvalues (or pseudospectrum) of non-self-adjoint semiclassical differential operators is due to the existence of quasimodes, that is, approximate local solutions to the eigenvalue problem. For scalar operators, the quasimodes appear generically since the bracket condition on the principal symbol is not satisfied for topological reasons.

In this paper we shall investigate how these results 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 quasisymmetrizable systems and systems of subelliptic type, for which we prove estimates on the resolvent.

pseudospectrum, spectral instability, system, semiclassical, non-self-adjoint, pseudodifferential operator
Mathematical Subject Classification 2000
Primary: 35S05
Secondary: 58J40, 47G30, 35P05
Received: 19 February 2008
Revised: 25 September 2008
Accepted: 22 October 2008
Published: 18 December 2008
Nils Dencker
Lund University
Centre for Mathematical Sciences
Box 118
SE-221 00 Lund