Fluctuation of spectra in random media (Q1820511)

We consider a bounded domain M in \({\mathbb{R}}^ 3\) with smooth boundary C. We put \(B(\epsilon;w)=\{x\in {\mathbb{R}}^ 3\); \(| x-w| <\epsilon \}\). Fix \(\beta\geq 1\). Let \(0<\mu_ 1(\epsilon;w(m))\leq \mu_ 2(\epsilon;w(m))\leq..\). be the eigenvalues of \(-\Delta\) in the set \(M_{\epsilon,w(m)}= M\setminus \cup^{\tilde m}_{t=1}B(\epsilon;w_ i^{(m)})\) 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 \(\tilde m\)-points. Let \(V(x)>0\) be \(C^ 0\)- function on M satisfying \(\int_{M}V(x)dx=1.\) We consider M as the probability space with the probability density \(V(x)\). Let \(M^{\tilde m}=\prod M\) be the probability space with the product measure. Fix \(\alpha >0\). Then, \(\mu_ j(\alpha /m;w(m))\) is a random variable on \(M^{\tilde m}\). Our aim is to know the precise asymptotic behaviour of \(\mu_ j(\alpha /m;w(m))\) as \(m\to \infty\). The following result which is a generalization of a result of \textit{R. Figari}, \textit{E. Orlandi} and \textit{S. Teta} [J. Stat. Phys. 41, 465-487 (1985)] is given: Theorem. Fix j. Assume \(\beta\in [1,12/11)\), \(V(x)=1/| M|\) (const.). Assume that the j-th eigenvalue \(\mu_ j\) of the Laplacian in M under the Dirichlet condition on C is simple. Then, the random variable \[ m^{1-(\beta /2)}((\mu_ j(\alpha /m;w(m))-(\mu_ j+4\pi \alpha m^{\beta -1}| M|^{-1}))) \] tends in distribution to Gaussian random variable \(\Pi_ j\) of mean 0 and variance \[ E(\Pi^ 2_ j)=4\pi \alpha (\int_{M}\phi_ j(x)^ 4| M|^{-1}dx-| M|^{-2}) \] as m tends to \(\infty\). Here \(\phi_ j\) is the normalized eigenfunction.