Nils Dencker, Lund,
Pseudospectra and subellipticity of Semiclassical Systems
Abstract:
The 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 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 false eigenvalues (or pseudospectrum) of
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
generically since the bracket condition on the principal symbol is not
satisfied for topological reasons.
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.