Pavel Exner and Sylwia Kondej
Bound states due to a strong $\delta$ interaction
supported by a curved surface
(63K, LaTeX)
ABSTRACT. We study the Schr\"odinger operator
$-\Delta -\alpha \delta (x-\Gamma )$ in $L^2(\R^3)$ with a
$\delta$ interaction supported by an infinite non-planar surface
$\Gamma$ which is smooth, admits a global normal parameterization
with a uniformly elliptic metric. We show that if $\Gamma $
is asymptotically planar in a suitable sense and $\alpha>0$ is
sufficiently large this operator has a non-empty discrete
spectrum and derive an asymptotic expansion of the eigenvalues in
terms of a ``two-dimensional'' comparison operator determined by
the geometry of the surface $\Gamma$. [A revised version, to
appear in J. Phys. A]