Spectral properties of random media
Shin Ozawa · Proceedings of the Japan Academy Series A Mathematical Sciences · 1984
This note is a continuation of our previous papers Ozawa [2], [3].We consider a bounded domain /2 in R with smooth boundary We put B(e;w)={xeR;Ix--wl}. Fix _1.Let0/(e;w(m)) _Z2(;w(m))___..be the eigenvalues of --z/(----div grad) in 9,) --9\?__B(;w)) under the Dirichlet condition on its boundary.Here h denotes the largest integer which does not exceed m , and w(m) denotes the set of h-points ) {w }__eLetV(x)>0 be C 1- class function on/2 satisfying V(x)dx 1.We consider/2 as the probability space with the probability density V(x)dx.Let /2= l-]=1/2 be the probability space with the product measure.