Remarks on the Cwikel-Lieb-Rozenblum and Lieb-Thirring estimates for Schrödinger operators on Riemannian manifolds
Let \(\Delta=\delta d\) be the Laplace operator on the \(L^2\) space of a complete Riemannian manifold. Consider the Schroedinger operator \(S:=\Delta+V\) where \(V\) is a potential function. Assume the positive part \(V_+\) of \(V\) is locally integrable and the negative part \(V_-\) is form-small. Let \(\{\lambda_1,\dots,\lambda_N\}\) be the eigenvalues of \(\Delta+V\) with negative real part. Let \(p(t,x,y)\) be the heat kernel of \(\Delta\). Let \(G\) be a non-trivial continuous convex function on \([0,\infty)\) which satisfies some suitable growth conditions and let \(g(1)\) be the Laplace transform at \(1\) of \(z^{-1}G(z)\). If the potential is real-valued and if \(\gamma>0\), the authors establish the estimate: \[ \sum_{j=1}^N(-\lambda_j)^\gamma\leq{\Gamma(\gamma+1)\over g(1)} \int_0^\infty {dt\over t^{\gamma+1}}\int_Mp(t,x,x)G(tV_-(x))dvol\,; \] there is a suitable inequality as well when \(\gamma=0\). More generally, if \(V\) is complex valued, then for \(\gamma\geq1\), the authors establish \[ \sum_{j=1}^N(-\Re(\lambda_j))^\gamma\leq {\Gamma(\gamma+1)\over g(1)} \int_0^\infty {dt\over t^{\gamma+1}}\int_Mp(t,x,x)G(t\Re(V)_-(x))dvol\,. \] The estimates in fact hold true for more general operators as they are based on an abstract result and an approximation argument that does not explicitly use the Laplacian.
scientific article; zbMATH DE number 5709155
scientific article