 09110 Mark S. Ashbaugh, Fritz Gesztesy, Marius Mitrea, and Gerald Teschl
 Spectral Theory for Perturbed Krein Laplacians in Nonsmooth Domains
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Jul 12, 09

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Abstract. We study spectral properties for $H_{K,\Omega}$, the Kreinvon Neumann
extension of the perturbed Laplacian $\Delta+V$ defined on
$C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in
a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of
nonsmooth domains which contains all convex domains, along with all domains
of class $C^{1,r}$, $r>1/2$. In particular, in the aforementioned context we
establish the Weyl asymptotic formula
\[
\#\{j\in\mathbb{N}\,\,\lambda_{K,\Omega,j}\leq\lambda\}
= (2\pi)^{n} v_n \Omega\,\lambda^{n/2}+O\big(\lambda^{(n(1/2))/2}\big)
\, \mbox{ as }\, \lambda\to\infty,
\]
where $v_n=\pi^{n/2}/ \Gamma((n/2)+1)$ denotes the volume of the unit ball
in $\mathbb{R}^n$, and $\lambda_{K,\Omega,j}$, $j\in\mathbb{N}$, are the
nonzero eigenvalues of $H_{K,\Omega}$, listed in increasing order
according to their multiplicities. We prove this formula by showing
that the perturbed Krein Laplacian (i.e., the Kreinvon Neumann extension of
$\Delta+V$ defined on $C^\infty_0(\Omega)$) is spectrally equivalent to the
buckling of a clamped plate problem, and using an abstract result of Kozlov
from the mid 1980's. Our work builds on that of Grubb in the early 1980's,
who has considered similar issues for elliptic operators in smooth domains,
and shows that the question posed by Alonso and Simon in 1980
pertaining to the validity of the above Weyl asymptotic formula
continues to have an affirmative answer in this nonsmooth setting.
We also study certain exteriortype domains $\Omega = \mathbb{R}^n\backslash K$,
$n\geq 3$, with $K\subset \mathbb{R}^n$ compact and vanishing Bessel capacity
$B_{2,2} (K) = 0$, to prove equality of Friedrichs and Krein Laplacians in
$L^2(\Omega; d^n x)$, that is, $\Delta_{C_0^\infty(\Omega)}$ has a unique
nonnegative selfadjoint extension in $L^2(\Omega; d^n x)$.
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