Small eigenvalues of Schrödinger operators over geometrically finite manifolds (Q6640723)

Let \(M\) be a a geometrically finite manifold of dimension \(m\) and sectional curvature \(K\) bounded by \(-1\leq K\leq -a^2<0\), for some \(0<a<1\). Let \(H\) be the universal covering space of \(M\), endowed with the lifted metric, and let \(\lambda_0=\lambda_0(H)\) be the bottom of the spectrum of the Laplace-Beltrami operator on \(H\).\N\NThe main result provides an upper bound for the number of eigenvalues of the Laplace-Beltrami operator of \(M\) in \([0,\lambda_0-\varepsilon]\) for any \(0<\varepsilon<\lambda_0\). This bound is a product of a value that depends on the geometry of \(M\) (the volume of the set of points of distance less than \(1\) to the convex core \(C\) of \(M\)) multiplied by a constant depending on \(m, a,\varepsilon\), but not on \(H\) or \(M\).\N\NFurthermore, the authors extend the above result for the case of Schrödinger operators on a vector bundle \(E\) over \(M\), whose pull-back to \(H\) is a bundle associated to a geometric structure, with structure group a covering group of \(\operatorname{SO}(m)\). Notably, the statement is very similar as the above one with minor changes depending on the bundle \(E\), but not on the corresponding potential.