We prove trace identities for commutators of operators, which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic Schrodinger operators and Schrodinger 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 case of a Schrodinger operator on an immersed manifold of dimension d, we derive a version of Reilly's inequality bounding the eigenvalue lambda(N+1) of the Laplace-Beltrami operator by a universal constant times vertical bar vertical bar h vertical bar N-2(infinity)2/d.