Johannes Sjöstrand

The asymptotic distribution of eigenvalues of self-adjoint differential operators in the high-energy limit, or the semi-classical limit, is a classical subject going back to H. Weyl of more than a century ago. In the last decades there has been a renewed interest in non-self-adjoint differential operators which have many subtle properties such as instability under small perturbations. Quite remarkably, when adding small random perturbations to such operators, the eigenvalues tend to distribute according to Weyl's law (quite differently from the distribution for the unperturbed operators in analytic cases). A first result in this direction was obtained by M. Hager in her thesis of 2005. Since then, further general results have been obtained, which are the main subject of the present book. Additional themes from the theory of non-self-adjoint operators are also treated. The methods are very much based on microlocal analysis and especially on pseudodifferential operators. The reader will find a broad field with plenty of open problems.

Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations

Bert-Wolfgang Schulze

Michael Demuth

Elmar Schrohe

Evolution equations describe many processes in science and engineering, and they form a central topic in mathematics. The first three contributions to this volume address parabolic evolutionary problems: The opening paper treats asymptotic solutions to singular parabolic problems with distribution and hyperfunction data. The theory of the asymptotic Laplace transform is developed in the second paper and is applied to semigroups generated by operators with large growth of the resolvent. An article follows on solutions by local operator methods of time-dependent singular problems in non-cylindrical domains. The next contribution addresses spectral properties of systems of pseudodifferential operators when the characteristic variety has a conical intersection. Bohr-Sommerfeld quantization rules and first order exponential asymptotics of the resonance widths are established under various semiclassical regimes. In the following article, the limiting absorption principle is proven for certain self-adjoint operators. Applications include Hamiltonians with magnetic fields, Dirac Hamiltonians, and the propagation of waves in inhomogeneous media. The final topic develops Hodge theory on manifolds with edges; its authors introduce a concept of elliptic complexes, prove a Hodge decomposition theorem, and study the asymptotics of harmonic forms.

Johannes Sjöstrand

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