Trace identities for commutators, with applications to the distribution of eigenvalues

TL;DR

This paper develops trace identities for operator commutators to establish universal bounds and inequalities for eigenvalues of Schrödinger operators and Laplace-Beltrami operators on manifolds, with applications in spectral geometry.

Contribution

It introduces new trace identities for commutators that lead to sharp eigenvalue bounds and monotonicity properties, extending classical inequalities in spectral theory.

Findings

Derived bounds on eigenvalues of Schrödinger operators.

Established universal monotonicity of eigenvalue moments.

Proved a geometric inequality relating eigenvalues to mean curvature.

Abstract

We prove trace identities for commutators of operators, which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic Schroedinger operators and Schroedinger operators on immersed manifolds. In particular, we prove bounds on the eigenvalue lambda_{N+1} in terms of the lower spectrum, bounds on ratios of means of eigenvalues, and universal monotonicity properties of eigenvalue moments, which imply sharp versions of Lieb-Thirring inequalities. In the geometric context we derive a version of Reilly's inequality, bounding the eigenvalue lambda_{N+1} of the Laplace-Beltrami operator on an immersed manifold of dimension d by a universal constant times the square of the maximal mean curvature times N^{2/d}.