Spectra of random media with many randomly distributed obstacles (Q1261610)

This paper is devoted to the eigenvalue problem of the Laplacian in \(\Omega_{w(m)}\) under the Dirichlet condition, where \(\Omega\) is a bounded domain in \(R^ 4\), \(\Omega_{w(m)}=\Omega\backslash\text{cl}\bigcup\{B(w_ i,m^{- 1}):i=1,2,\dots,n\}\), \(w(m)=(w_ 1,\dots,w_ n)\in\Omega^ n\), \(n=[m^ \beta]\), and \(\beta\in[2,12/5)\). Suppose \(\Omega^ n\) is endowed with the product probability measure defined by a positive continuous density \(V\) on \(\text{cl} \Omega\). Eigenvalues of the operator \(-\Delta\) in \(\Omega_{w(m)}\) are viewed as random variables on \(\Omega^ n\). The author proves that if \(m\to\infty\), then the spectra of \(-\Delta\) in \(\Omega_{w(m)}\) under the Dirichlet condition on \(\partial\Omega_{w(m)}\) tend in probability to the spectra of the Schrödinger operator \(-\Delta+cV\) in \(\Omega\) under the Dirichlet condition on \(\partial\Omega\), and gives the remainder estimation. In the proof he uses the perturbative expansion of the Green function.