Spectral theory of some degenerate elliptic operators with local singularities

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 b 2 ○ p ( x , D ) ○ b 1 where b i ∈ L r i ( R n , w i ) , w i ∈ A ∞ , i = 1 , 2 , and p ( x , D ) ∈ Ψ 1 , 0 − ϰ , ϰ > 0 . Finally we deal with the ‘negative spectrum’ of some operator H γ = A − γ V 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 > 0 , self-adjoint in L 2 ( R n ) . 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.