Abstract
Let A=A 0+v(x) where A 0 is a second-order uniformly elliptic self-adjoint operator in R d and v is a real valued polynomially growing potential. Assuming that v and the coefficients of A 0 are Hölder 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.
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Zielinski, L. Asymptotic Distribution of Eigenvalues for a Class of Second-Order Elliptic Operators with Irregular Coefficients in R d . Mathematical Physics, Analysis and Geometry 5, 145–182 (2002). https://doi.org/10.1023/A:1016286227961
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DOI: https://doi.org/10.1023/A:1016286227961