On an Analogue of the Rozenblum-lieb-cwikel Inequality for the Biharmonic Operator on a Riemannian Manifold
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چکیده
0. Introduction Let ∆ be the Laplacian on R, l > 0 an integer and V ≥ 0 a measurable function (“weight-function”). Consider the eigenvalue problem (0.1) λ(−∆)u = V u. The following result was proved by Rosenblum [Roz]: Theorem 0.1. Let 2l < d and V ∈ L d 2l (R). Then the non-zero spectrum of the problem (0.1) consists of positive eigenvalues λk (counted according to their multiplicities), and for their distribution function n(λ) = #{k : λk > λ}, λ > 0, the following estimate and asymptotics hold: (0.2) n(λ) ≤ C(d, l)λ−d/2l ∫
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تاریخ انتشار 2004