Spectral properties of random media (Q1069561)

We consider a bounded domain M in \(R^ 3\) with smooth boundary \(\partial M\). We put \(B(\epsilon;w)=\{x\in R^ 3;| x-w| <\epsilon \}.\) Fix \(\beta\geq 1\). Let \(0<\mu_ 1(\epsilon;w(m))\leq \mu_ 2(\epsilon;w(m))<...\) be the eigenvalues of -\(\Delta\) \((=-div \text{grad})\) in \(M_{\epsilon,w(m)}=M\setminus \cup^{\tilde m}_{i=1}B(\epsilon;w_ i^{(m)}).\) Here \(\tilde m\) denotes the largest integer which does not exceed \(m^{\beta}\), and w(m) denotes the set of \(\tilde m-\)points \(\{w_ i^{(m)}\}^{\tilde m}_{i=1}\in M^{\tilde m}\). Let \(V(x)>0\) be a \(C^ 1\) class function on M satisfying \(\int_{M}V(x)dx=1\). We consider \(M^{\tilde m}\) as the probability space with the product measure where M is the probability space with the probability density V(x)dx. Theorem. Fix \(\beta\in [1,3)\). Assume that \(V(x)>0\). Fix \(\epsilon >0\) and k. Then, there exists a constant \(\delta (\beta)>0\) independent of m such that \[ \lim P(w(m)\in M^{\tilde m};\quad m^{\delta '-(\beta - 1)}| \mu_ k(\alpha /m;w(m))-\mu_{k,m}| <\epsilon)=1 \] holds for any \(\epsilon >0\) and \(\delta\) '\(\in [0,\delta (\beta))\). Here \(\mu_{k,m}\) denotes the k-th eigenvalue of \(-\Delta +4\pi \alpha m^{\beta -1}V(x)\) in M under the Dirichlet condition on \(\partial M.\) For recent progress on related topics, see the author, ''Fluctuation of spectra in random media.'' in: K. Ito and N. Ikeda (eds.): Probabilistic methods in mathematical physics, Proc. Taniguchi Int. Symp. Probabilistic Methods Math. Physics. (1985).