Viscosity limits for 0th order pseudodifferential operators

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 $P + i νΔ$, $ν> 0$. By adapting the Helffer--Sjöstrand theory of scattering resonances we show that in a complex neighbourhood of the continuous spectrum eigenvalues of $P + i νΔ$ have limits as viscosity $ν$ goes to 0. In the simplified setting of tori this justifies claims made in the physics literature.

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