Commutators, spectral trace identities, and universal estimates for eigenvalues

By M. Levitin, L. Parnovski. 2001

64 citations recorded by the source corpus.

Using simple commutator relations, we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self-adjoint operator acting in a Hilbert space. Applications involve abstract universal estimates for the eigenvalue gaps. As particular examples, we present simple proofs of the classical universal estimates for eigenvalues of the Dirichlet Laplacian, as well as of some known and new results for other differential operators and systems. We also suggest an extension of the methods to the case of non-self-adjoint operators.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/62d61a947de7b78c215411946dcf40efac81d0d7

How researchers see this paper

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In Computational Theory and Mathematics

1. Hongcang Yang
2. Qing Cheng
3. R. Benguria
4. M. Ashbaugh
5. Daniel Daners
6. C. Xia
7. Ahmad El Soufi
8. S. Ilias

In Mathematical Physics

1. Hongcang Yang
2. Qing Cheng
3. R. Benguria
4. M. Ashbaugh
5. Daniel Daners

Citations in the corpus

• Isoperimetric Inequalities for Eigenvalues of the Laplacian (2007, methodology): “Then the oscillation frequencies of the drum are given by the eigenvalues of the Laplace operator −∆ΩD on Ω with Dirichlet boundary conditions, up to a constant that depends on the drum’s tension and mass density.”
• Lower order eigenvalues of Dirichlet Laplacian (2008, background): “We believe that it is not possible to derive our present results from [ 24 ], at least if the ambient Riemannian manifold has non-constant curvature.”