On an elaboration of M. Kac's theorem concerning eigenvalues of the Laplacian in a region with randomly distributed small obstacles
We consider a bounded domain \(\Omega\) in \({\mathbb{R}}^ 3\) with smooth boundary \(\Gamma\). We put \(B(\epsilon;w)=\{x\in {\mathbb{R}}^ 3\quad:\quad | x-w|<\epsilon \}.\) Let \(0<\mu_ 1(\epsilon;w(m))\leq \mu_ 2(\epsilon;w(m))\leq...\) be the eigenvalues of \(-\Delta(=-div \text{grad})\) in \(\Omega_{\epsilon,w(m)}=\Omega \backslash \overline{\cup^{m}_{i=1}B(\epsilon;w_ i^{(m)})}\) under the Dirichlet condition on its boundary. Here w(m) denotes the set of m points \(\{w_ i^{(m)}\}^ m_{i=1}\in \Omega^ m\). Let V(x)\(\geq 0\) be a \(C^ 1\)-class function on \({\bar \Omega}\) satisfying \(\int_{\Omega}V(x)\quad dx=1.\) We consider \(\Omega\) as the probability space with the probability density V(x) dx. Let \(\Omega^ m=\prod^{m}_{i=1}\Omega\) be the probability space with the product measure. We prove the Theorem: Fix \(\alpha >0\) and k. Then \[ \lim_{m\to \infty}P(w(m)\in \Omega^ m;\quad m^{\delta}| \mu_ k(\alpha /m;w(m))-\mu^ V_ k|<\epsilon)=1 \] holds for any \(\epsilon>0\) and \(\delta \in [0,1/4)\). Here \(\mu^ V_ k\) denotes the kth eigenvalue of \(-\Delta +4\pi \alpha V(x)\) in \(\Omega\) under the Dirichlet condition on \(\Gamma\). [See also the author's paper in Proc. Jap. Acad., Ser. A 60, 43-45 (1984)].
scientific article; zbMATH DE number 3859587