Viscosity limits for 0th order pseudodifferential operators

Abstract

Motivated by the work of Colin de Verdière and Saint-Raymond on spectral theory 0th order pseudodifferential operators on tori we consider viscosity limits in which 0th order operators $P$ are replaced by $P+i\nu \Delta$, $\nu >0$. By adapting the Helffer--Sjöstrand theory of scattering resonances we show that in a complex neighbourhood of the continuous spectrum eigenvalues of $P+i\nu \Delta$ have limits as viscosity $\nu$ goes to 0. In the simplified setting of tori this justifies claims made in the physics literature.

Proposition 6.3 added and some small issues corrected