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

2009/03/03 by Harrell, Evans M., Stubbbe, Joachim
#Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.0903.0563

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.

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