09-110 Mark S. Ashbaugh, Fritz Gesztesy, Marius Mitrea, and Gerald Teschl

Spectral Theory for Perturbed Krein Laplacians in Nonsmooth Domains (296K, LaTex2e) Jul 12, 09

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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 $L^2(\Omega; d^n x)$.

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