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.

Schrödinger operators and their spectral properties play an important role in the mathematical description of quantum mechanical systems. In many situations one uses idealized models with singular potentials (in particular, delta-potentials supported on curves, surfaces, and manifolds), as these are mathematically more accessible and can be solved explicitely in some cases. In this project an abstract operator theoretic approach to singular perturbations of selfadjoint operators in Hilbert spaces was developed. Here selfadjoint operators are interpreted as extensions of suitable underlying symmetric operators and are parametrized with abstract boundary conditions. It was shown that the spectral properties (isolated and embedded eigenvalues, continuous, absolutely continuous and singular continuous spectrum) of the perturbed operator can be completely described with an abstract Titchmarsh-Weyl m-function. With the help of Krein type formulae the singular values and Schattenvon Neumann properties of the differences of the re- solvents of the perturbed and unperturbed operator were investigated. The important case of trace class perturbations was treated separately and an explicit representation formula for the scattering matrix in terms of the abstract Titchmarsh-Weyl m-function was found; this relation is a natural generalization of a classical result for finite rank perturbations. Furthermore, the Krein spectral shift function was also expressed in terms of the abstract Titchmarsh-Weyl m-function. The abstract results in this project were applied in various situations to singular perturbations of Schrödinger operators. In particular, the existence of eigenvalues, eigenvalue bounds, asymptotic spectral properties, as well as trace formulas and scattering problems for d and d -potentials supported on bounded and unbounded hypersurfaces in n- dimensionalen Euclidean space were discussed. It also turned out that there is an unexpected connection between Schrödinger opera- tors with d and d -potentials on Lipschitz partitions and the chromatic number of the partition. As an explicit model d-perturbations supported on closed curves in threedimensional space were systematically studied and relations between the geometric properties of the curve and the spectrum of the corresponding Schrödinger operators were proved. Among other closely related topics that were discussed in this project are singular perturbations of Dirac operators, elliptic operators with variable coefficients, a class of inverse spectral problems of Calderon or Gelfand type, non-selfadjoint operators with general boundary conditions, Schrödinger operators with complex potentials, non-elliptic differential operators from the mathematical theory of metamaterials, abstract meromorphic operator functions, as well as perturbation problems and eigenvalue bounds for operators in indefinite inner product spaces.