P. Exner, M. Fraas
On geometric perturbations of critical Schr\"odinger
operators with a surface interaction
(42K, LaTeX 2e)
ABSTRACT. We study singular Schr\"odinger operators with an
attractive interaction supported by a closed smooth surface $\man
A\subset\mathbb{R}^3$ and analyze their behavior in the vicinity
of the critical situation where such an operator has empty
discrete spectrum and a threshold resonance. In particular, we
show that if $\man A$ is a sphere and the critical coupling is
constant over it, any sufficiently small smooth area preserving
radial deformation
gives rise to isolated eigenvalues. On the other hand, the
discrete spectrum may be empty for general deformations. We also
derive a related inequality for capacities associated with such
surfaces.