Semiklassische Analysis mit Singularitäten

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In this project we are going to study the construction of semiclassical spectral asymptotics for pseudodifferential operators using singular invariant varieties of the corresponding classical Hamiltonian. For a wide class of pseudodifferential operators, we plan to study the existence of such asymptotics, the accuracy of the approximation, the relationship between semiclassical asymptotics and the topological classification of critical manifolds, and various related questions. We expect that the results will be applicable to the asymptotic description of the spectral bands in periodic and gauge-periodic problems and to finding spectral asymptotics in problems with non-integrable Hamiltonians.