Viscosity Limits for 0th order operators
Recent developments in microlocal analysis October 14, 2019 - October 18, 2019
Location: MSRI: Simons Auditorium
35A27 - Microlocal methods; methods of sheaf theory and homological algebra in PDE [See also 32C38, 58J15]
For self-adjoint pseudodifferential operators of order 0, Colin de Verdiere and Saint-Raymond introduced natural dynamical conditions (motivated by the study of internal waves in fluids) guaranteeing absolute continuity of the spectrum. I will present an alternative approach to obtaining such results based on Melrose's radial propagation estimates from scattering theory (joint work with S Dyatlov). I will then explain how an adaptation of the Helffer–Sjoestrand theory of scattering resonances shows that in a complex neighbourhood of the continuous spectrum viscosity eigenvalues have limits as viscosity goes to 0. Here the viscosity eigenvalues are the eigen- values of the original operator to which an anti-self-adjoint elliptic 2nd order operator is added (joint work with J Galkowski).
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