The Rozenblum-Lieb-Cwikel inequality for Markov generators
Let \(N(V)\) be the number of negative eigenvalues of the Schrödinger operator \(H_0+V\), \(H_0 = -\Delta\) in \(L^2({\mathbb R}^d)\). According to the well-known Rozenblum-Lieb-Cwikel inequality \[ N(V) \leq \int V_{+}^{d/2} dx, \] where \(V_{+} = (| V| +V)/2\). Several proofs of this inequality due to Rozenblum, Lieb, Cwikel, Li-Yau, Conlon are known. The authors show that the approach of Li and Yau works in the more general situation, when \(H_0\) is a Markov generator in \(L^2(\Omega,\sigma)\), where \((\Omega,\sigma)\) is a measure space with sigma-finite measure. Several applications of this result are included. In particular, the cases of Riemannian manifolds and hyperbolic spaces are discussed.
scientific article; zbMATH DE number 1038445
scientific article