Evolution Equations, Feshbach Resonances, Singular Hodge Theory

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