Mark S. Ashbaugh, Fritz Gesztesy, Marius Mitrea, and Gerald Teschl
Spectral Theory for Perturbed Krein Laplacians in Nonsmooth Domains
(296K, LaTex2e)

ABSTRACT.  We study spectral properties for $H_{K,\Omega}$, the Krein--von 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 
non-zero 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 Krein--von 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 exterior-type 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 self-adjoint extension in $L^2(\Omega; d^n x)$.