A lower bound for the number of negative eigenvalues of Schrödinger operators. (English) Zbl 1356.53044

Grigor’yan, Alexander; Nadirashvili, Nikolai; Sire, Yannick

J. Differ. Geom. 102, No. 3, 395-408 (2016).

It is a classical and important problem to give bounds on the eigenvalues of the Laplace operator of a compact Riemannian manifold \(M\), and more generally the Schrödinger operators on a compact Riemannian manifold \(M\). For every potential function \(V\), the Schrödinger operator \(-\Delta-V\) has only finitely many negative eigenvalues, where the Laplace operator \(-\Delta\) is nonnegative. A basic problem is to give bounds on the number of such negative eigenvalues. There have been a lot of work on the upper bounds. In this paper, the authors give a lower bound for the number of negative eigenvalues in terms of the integral of the potential \(V\). This generalizes the earlier result when the potential \(V\) is nonnegative. The idea of their proof is to reduce the general case to this special case by considering a certain variational problem for \(V\).

Reviewer: Lizhen Ji (Ann Arbor)

MSC:

53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J50	Spectral problems; spectral geometry; scattering theory on manifolds
35P15	Estimates of eigenvalues in context of PDEs

Keywords:

negative eigenvalue; Schrödinger operator