Reduced Weyl asymptotics for pseudodifferential operators on bounded domains I. The finite group case

Abstract

Let G subset of O(n) be a compact group of isometrics acting on n-dimensional Euclidean space R-n, and X a bounded dornain in R-n which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A(0) in L-2(R-n ) with G-invariant Weyl symbol, and assume that it is semi-bounded from below. We show that the spectrum of the Friedrichs extension A of the operator res o A(0) o ext : C-c(infinity)(X) -> L-2(X) is discrete, and derive asymptotics for the number N-chi(lambda) of eigenvalues of C A less or equal; and with eigenfunctions in the X-isotypic component of L-2(X) as lambda -> infinity, giving also an estimate for the remainder term in case that G is a finite group. In particular, we show that the multiplicity of each unitary it-reducible representation in L-2(X) is asymptotically proportional to its dimension. (c) 2008 Elsevier Inc. All rights reserved.