Discrete Spectra of Self-adjoint Operators

  • Konrad Schmüdgen
Part of the Graduate Texts in Mathematics book series (GTM, volume 265)


In Chap. 12, we investigate discrete spectra of self-adjoint operators. The first main result is the Courant–Fischer–Weyl min–max principle. We state two versions of this theorem, one for operator domains and one for form domains. It allows us to compare the eigenvalues and discrete spectra of lower semibounded self-adjoint operators. Some applications of the min–max principle (Rayleigh–Ritz method, Schrödinger operators) and Temple’s inequality are briefly discussed. Another section contains some results concerning the existence of positive or negative eigenvalues of self-adjoint operators and Schrödinger operators. In the final section, Weyl’s classical asymptotic formula for the eigenvalues of the Dirichlet Laplacian on a bounded open Jordan measurable subset of ℝ d is proved.


Discrete Spectrum Negative Eigenvalue Ritz Method Theorem XIII Heinz Inequality 
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Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • Konrad Schmüdgen
    • 1
  1. 1.Dept. of MathematicsUniversity of LeipzigLeipzigGermany

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