Pavel Exner
Spectral Properties of Schroedinger operators with a strongly
attractive $\delta$ interaction supported by a surface
(49K, LaTeX)
ABSTRACT. We investigate the operator $-\Delta -\alpha \delta (x-\Gamma)$ in
$L^2(\mathbb{R}^3)$, where $\Gamma$ is a smooth surface which is
either compact or periodic and satisfies suitable regularity
requirements. We find an asymptotic expansion for the lower part
of the spectrum as $\alpha\to\infty$ which involves a
``two-dimensional'' comparison operator determined by the geometry
of the surface $\Gamma$. In the compact case the asymptotics
concerns negative eigenvalues, in the periodic case Floquet
eigenvalues. We also give a bandwidth estimate in the case when a
periodic $\Gamma$ decomposes into compact connected components.
Finally, we comment on analogous systems of lower dimension and
other aspects of the problem.