SOSP - Schrödinger operators and singular perturbations

The analysis and spectral theory of partial differential operators has developed further and advanced versatilely during the last years. One of the reasons for this is that modern techniques from abstract operator theory have been applied successfully to PDE problems. In particular, recent concepts from extension theory of symmetric operators were applied to Schrödinger operators with delta point potentials and yield deeper insights into their spectral properties. In the first part of this project an abstract approach to singular perturbations of selfadjoint operators in Hilbert spaces will be developed and afterwards, in the second part, this method will be applied to Schrödinger operators with general delta potentials supported on curves, surfaces, and manifolds. In the first, abstract part of the project singularly perturbed selfadjoint operators will be considered as extensions of an underlying unperturbed symmetric operator. With the help of so-called boundary triples and their corresponding abstract Weyl functions the selfadjoint extensions will be parametrized and their spectral properties will be analyzed in detail. The construction of the boundary triples and the analytic properties of the corresponding Weyl functions depend on the degree of singularity of the perturbations. With the help of a Krein type formula it will be investigated under which conditions on the perturbations the resolvent differences of the perturbed and unperturbed operators belong to some Schatten-von-Neumann ideals. In particular the trace class case, which is important for mathematical scattering theory, is included here. In the second part of the project these abstract results will be applied to Schrödinger operators with weighted delta potentials (and their distributional derivatives) supported on curves, surfaces, and manifolds. With more explicit representations of the Weyl functions the effects of the perturbations on the spectra and certain corresponding inverse problems will be investigated. Depending on the regularity and dimension of the manifolds as well as the degree of the derivatives of the potentials different methods developed in the abstract part will be applied here. Many of the results remain true for magnetic Schrödinger operators and more general elliptic differential operators with variable coefficients.