Universal bounds and semiclassical estimates for eigenvalues of abstract Schroedinger operators

2008/08/08 by Evans M. Harrell II, Harrell, Evans M., Joachim Stubbe +1
Mathematics · #35J10 #35J25 #81Q10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP) #math.AP #math.SP #msc:35J10 #msc:35J25 #msc:81Q10

paper · pdf · doi:10.48550/arxiv.0808.1133

arxiv created 2008/08/08 · arxiv updated 2009/12/01

Abstract

We prove trace inequalities for a self-adjoint operator on an abstract Hilbert space. These inequalities lead to universal bounds on spectral gaps and on moments of eigenvalues lambda_k that are analogous to those known for Schroedinger operators and the Dirichlet Laplacian, on which the operators of interest are modeled. In addition we produce inequalities that are new even in the model case. These include a family of differential inequalities for generalized Riesz means and theorems stating that arithmetic means of lambda_k^p for p <= 3 are universally bounded from above by multiples of the geometric mean of the lambda_k. For Schroedinger operators and the Dirichlet Laplacian these bounds are Weyl-sharp, i.e., saturated by the standard semiclassical estimates for lambda_k at large k.

Related

• Trace identities for commutators, with applications to the distribution of eigenvalues
• Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds,
• Counting Schrödinger boundstates: semiclassics and beyond
• Effective eigenvalue approximation from moments for self-adjoint trace-class operators
• Universal Monotonicity of Eigenvalue Moments and Sharp Lieb-Thirring Inequalities