Asymptotic Distribution of Eigenvalues for a Class of Second-Order Elliptic Operators with Irregular Coefficients in R d

Abstract

Let A=A0+v(x) where A0 is a second-order uniformly elliptic self-adjoint operator in Rd and v is a real valued polynomially growing potential. Assuming that v and the coefficients of A0 are Holder continuous, we study the asymptotic behaviour of the counting function N(A,λ) (λ→∞) with the remainder estimates depending on the regularity hypotheses. Our strongest regularity hypotheses involve Lipschitz continuity and give the remainder estimate N(A,λ)O({λ}−μ), where μ may take an arbitrary value strictly smaller than the best possible value known in the smooth case. In particular, our results are obtained without any hypothesis on critical points of the potential.