The analysis and spectral theory of Schrödinger operators with δ-potentials have attracted an
enormous attention in the last decades. This field in mathematical physics is particularly challenging,
as it crosses boundaries between operator theory, mathematical analysis and partial
differential equations. Point interaction models (also called zero range or δ-interaction models)
serve as rough approximations of more complicated interactions in quantum systems, and
are regarded as solvable models in the sense that the spectrum, eigenfunctions, resonances and
scattering quantities can be computed explicitly.
This project makes a new step towards more realistic models and goes far beyond the
present state of the art in the theory of singular and supersingular δ-interaction models. Instead
of δ-interactions supported on finite (or discrete) point sets we will investigate δ-interactions
which are supported on general curves, surfaces and manifolds of arbitrary co-dimension. This
requires advanced analytic tools, in particular, refined methods from operator theory and partial
differential equations. Our main objective is to develop an abstract perturbation approach to infinite
dimensional singular perturbations of selfadjoint operators, and to provide a thorough and
in-depth spectral analysis of Schrödinger operators and more general elliptic partial differential
operators with singular and supersingular δ-potentials supported on manifolds. Furthermore,
it is planned to investigate closely related scattering and multidimensional inverse problems,
to analyse bound states of Schrödinger operators with δ-interactions on curves and special surfaces,
and to discuss some explicitly solvable models. The proposed project is of fundamental
nature since only certain cases of δ-perturbations supported on curves and manifolds have been
understood by the time, and a general operator theoretic approach and a comprehensive spectral
analysis of singular and supersingular δ-interactions – as is our vision – does not exist until
now.