Boundary Value Problems, Schrödinger Operators, Deformation Quantization: Advances in Partial Differential Equations

The analysis of boundary value problems has a long tradition in mathematics. Understanding the criteria for solvability and the structure of the solutions is of central interest both for theory and applications. Boundary value problems on manifolds with singularities present an additional challenge. They exhibit a wealth of analytic and algebraic structures, also under the aspect of index theory. In the first contribution to this volume, boundary value problems without the transmission condition are interpreted as particular problems on manifolds with edges; it deals with the new effects caused by variable and branching asymptotics. In the second paper, a pseudo-differential calculus is constructed for boundary value problems on manifolds with conical
singularities. A concept of ellipticity is introduced that allows a parametrix construction and entails the Fredholm property in weighted Sobolev spaces. Moreover, this approach lays the foundations for treating boundary value problems on manifolds with edges. Two further contributions deal with deformation quantization, an important topic of Mathematical Physics. The first one gives a complete proof of the index theorem in deformation quantization, while the other one treats trace densities. The final article in this volume, also from the area of Mathematical Physics, presents new results on the spectrum of perturbed periodic Schrödinger operators.