Trace Identities for Commutators, with Applications to the Distribution of Eigenvalues

We prove trace identities for commutators of operators, which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic Schrödinger operators and Schrödinger operators on immersed manifolds. In particular, we prove bounds on the eigenvalue λN+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 λN+1 of the Laplace-Beltrami operator on an immersed manifold of dimension d by a universal constant times ‖h‖∞N.