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

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 $$V$$.

### 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
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