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Motivated by the work of Colin de Verdi\`ere and Saint-Raymond on spectraltheory 0th order pseudodifferential operators on tori we consider viscositylimits in which 0th order operators $ P $ are replaced by $ P + i \nu \Delta $,$ \nu > 0 $. By adapting the Helffer--Sj\"ostrand theory of scatteringresonances we show that in a complex neighbourhood of the continuous spectrumeigenvalues 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 physicsliterature.

Viscosity limits for 0th order pseudodifferential operators