Reduced Weyl asymptotics for pseudodifferential operators on bounded domains II. The compact group case

2007/10/01 by Cassanas, Roch, Ramacher, Pablo
#20C99 #35P20 #47G30 #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.0710.0126

Abstract

Let G⊂ Ø(n) be a compact group of isometries acting on n-dimensional Euclidean space \Rⁿ, and \bfX a bounded domain in \Rⁿ which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A₀ in Ł²(\Rⁿ) that commutes with the regular representation of G, and assume that it is elliptic on \bfX. We show that the spectrum of the Friedrichs extension A of the operator res ∘ A₀ ∘ ext: \CT(\bfX) → Ł²(\bfX) is discrete, and using the method of the stationary phase, we derive asymptotics for the number N_χ(λ) of eigenvalues of A equal or less than λ and with eigenfunctions in the χ-isotypic component of Ł²(\bfX) as λ→ ∞, giving also an estimate for the remainder term for singular group actions. Since the considered critical set is a singular variety, we recur to partial desingularization in order to apply the stationary phase theorem.

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