Commutators, spectral trace identities, and universal estimates for eigenvalues

Michael Levitin, Leonid Parnovski

Research output: Contribution to journalArticlepeer-review

40 Citations (Scopus)


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. © 2002 Elsevier Science (USA).

Original languageEnglish
Pages (from-to)425-445
Number of pages21
JournalJournal of Functional Analysis
Issue number2
Publication statusPublished - 10 Jul 2002


  • Commutator identities
  • Dirichlet eigenvalues
  • Eigenvalues estimates
  • Elasticity
  • Hile-Protter inequality
  • Laplace operator
  • Neumann eigenvalues
  • Payne-Pólya-Weinberger inequalities
  • Schrödinger operator
  • Spectral gap
  • Thomas-Reiche-Kuhn sum rule
  • Yang inequalities


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