Bert-Wolfgang Schulze

Michael Demuth

Elmar Schrohe

Johannes Sjöstrand

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.

Advances in Partial Differential Equations

Evolution Equations, Feshbach Resonances, Singular Hodge Theory

Sergio Albeverio

This volume focuses on recent developments in non-linear and hyperbolic equations. It will be a most valuable resource for researchers in applied mathematics, the theory of wavelets, and in mathematical and theoretical physics. Nine up-to-date contributions have been written on invitation by experts in the respective fields. The book is the third volume of the subseries "Advances in Partial Differential Equations".

A Volume of Advances in Partial Differential Equations

Nonlinear Hyperbolic Equations, Spectral Theory, and Wavelet Transformations

Partial differential equations constitute an integral part of mathematics. They lie at the interface of areas as diverse as differential geometry, functional analysis, or the theory of Lie groups and have numerous applications in the applied sciences. A wealth of methods has been devised for their analysis. Over the past decades, operator algebras in connection with ideas and structures from geometry, topology, and theoretical physics have contributed a large variety of particularly useful tools. One typical example is the analysis on singular configurations, where elliptic equations have been studied successfully within the framework of operator algebras with symbolic structures adapted to the geometry of the underlying space. More recently, these techniques have proven to be useful also for studying parabolic and hyperbolic equations. Moreover, it turned out that many seemingly smooth, noncompact situations can be handled with the ideas from singular analysis. The three papers at the beginning of this volume highlight this aspect. They deal with parabolic equations, a topic relevant for many applications. The first article prepares the ground by presenting a calculus for pseudo differential operators with an anisotropic analytic parameter. In the subsequent paper, an algebra of Mellin operators on the infinite space-time cylinder is constructed. It is shown how timelike infinity can be treated as a conical singularity.

Elmar Schrohe

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