Abstract:
In this lecture we define self-adjoint Schrödinger operators with
$\delta$ and $\delta'$-potentials
supported on a smooth compact hypersurface explicitly via boundary
conditions.
Some qualitative spectral properties of these operators are investigated
and a variant of Krein's
formula is shown. Furthermore, Schatten-von Neumann type estimates for
the differences of
the powers of the resolvents of the Schrödinger operators with $\delta$
and $\delta'$-potentials,
and the Schrödinger operator without a singular interaction are
discussed. An immediate consequence
of these estimates is the existence and completeness of the wave
operators of the corresponding
scattering systems, as well as the unitary equivalence of the absolutely
continuous parts of the spectra
of the singularly perturbed and unperturbed Schrödinger operators.
The talk is based on joint work with Matthias Langer (University of
Strathclyde, Glasgow) and
Vladimir Lotoreichik (Graz University of Technology).
