J. Behrndt, M. Langer and V. Lotoreichik,
Schrödinger operators with δ and δ'-potentials supported on hypersurfaces,
Ann. Henri Poincaré 14 (2013), 385–423
Abstract:
Self-adjoint Schrödinger operators with δ and δ′-potentials
supported on a smooth compact hypersurface are defined explicitly via boundary conditions.
The spectral properties of these operators are investigated, regularity results on the
functions in their domains are obtained, and analogues of the
Birman–Schwinger principle and a variant of Krein’s formula are shown.
Furthermore, Schatten–von Neumann type estimates for the differences
of the powers of the resolvents of the Schrödinger operators
with δ and δ′-potentials, and the Schrödinger operator
without a singular interaction are proved.
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 singularly
perturbed and unperturbed Schrödinger operators.
In the proofs of our main theorems we make use of abstract methods
from extension theory of symmetric operators, some algebraic considerations
and results on elliptic regularity.