  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}.
