Spectral theory of some degenerate elliptic operators with local singularities

Abstract

This paper is based on our previous results (Haroske and Skrzypczak (2008) [23], Haroske and Skrzypczak (in press) [25]) on compact embeddings of Muckenhoupt weighted function spaces of Besov and Triebel-Lizorkin type with example weights of polynomial growth near infinity and near some local singularity. Our approach also extends (Haroske and Triebel (1994) [21]) in various ways. We obtain eigenvalue estimates of degenerate pseudodifferential operators of type b2o p(x,D)ob1 where bi ∈ Lri(Rn,wi), wi ∈ A∞, i = 1,2, and p(x,D) ∈ ψ1,0-κ{script}, κ{script} > 0. Finally we deal with the 'negative spectrum' of some operator Hγ = A - γV for γ → ∞, where the potential V may have singularities (in terms of Muckenhoupt weights), and A is a positive elliptic pseudodifferential operator of order κ{script} > 0, self-adjoint in L2(Rn). This part essentially relies on the Birman-Schwinger principle. We conclude this paper with a number of examples, also comparing our results with preceding ones. © 2010 Elsevier Inc.