Viscosity limits for 0th order pseudodifferential operators

Motivated by the work of Colin de Verdi\`ere 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--Sj\"ostrand 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.