Exponentially sharp spectrum of non-selfadjoint 1D operators via analytic microlocal analysis

I will report on recent advances concerning the computation of spectra of non-selfadjoint semiclassical 1D (pseudo)differential or Toeplitz operators with exponentially small errors. The general idea, based on the well-known Bohr-Sommerfeld rule in the selfadjoint case, is to use a bit of geometry to analyse classical Hamiltonians in the complexified phase space. At the quantum level, this leads to complex Fourier integral operators via Sjöstrand’s theory. I will present applications to non-selfadjoint perturbations of self-adjoint operators, and open questions related to the non-perturbative regime. A large part of the talk will be based on results by Duraffour and Reguer.