Random media and eigenvalues of the Laplacian
We consider a bounded domain \(\Omega\) in \(R^ 3\) with smooth boundary \(\Gamma\). We put \(B(\epsilon;w)=\{x\in R^ 3;\quad | x-w| <\epsilon \}.\) Fix \(\beta\geq 1\). Let \(0<\mu_ 1(\epsilon;w(m))\leq \mu_ 2(\epsilon;w(m))\leq...\) be the eigenvalues of -\(\Delta\) in \(\Omega_{\epsilon,w(m)}=\Omega \setminus \cup^{\tilde m}_{i=1}B(\epsilon;w_ i^{(m)})\) under the Dirichlet condition on its boundary. Here \(\tilde m\) denotes the largest integer which does not exceed \(m^{\beta}\), and w(m) denotes the set of m-points \(\{w_ i^{(m)}\}^{\tilde m}_{i=1}\in \Omega^{\tilde m}.\) Let \(V(x)>0\) be a \(C^ 1\)-class function on \({\bar \Omega}\) satisfying \(\int_{\Omega}V(x)dx=1\). We consider \(\Omega\) as the probability space with the probability density V(x)dx. Let \(\Omega^{\tilde m}=\prod^{\tilde m}_{i=1}\Omega\) be the probability space with the product measure. Theorem: Assume that \(1\leq \beta <9/8\) and \(V(x)>0\). Fix \(\alpha >0\) and k. Then, there exists a constant \(\delta (\beta)>0\) independent of m such that \[ \lim_{m\to \infty}{\mathbb{P}}(w(m)\in \Omega^{\tilde m};\quad m^{\delta '-(\beta -1)}| \mu_ k(\alpha /m;w(m))-\mu^ V_{k,m}| <\epsilon)=1 \] holds for any \(\epsilon >0\) and \(\delta '\in [0,\delta (\beta)).\) Here \(\mu^ V_{k,m}\) denotes the k-th eigenvalue of \(-\Delta +4\pi \alpha m^{\beta -1}V(x)\) in \(\Omega\) under the Dirichlet condition on \(\Gamma\).
scientific article; zbMATH DE number 3884653
scientific article