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Lower bounds for eigenvalues of self-adjoint problems.

G G Gundersen1

  • 1Mathematics Department, University of New Orleans, New Orleans, Louisiana 70122.

Proceedings of the National Academy of Sciences of the United States of America
|November 1, 1979
PubMed
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Researchers derived a new inequality for discrete eigenvalues in differential equations. This method provides explicit lower bounds for eigenvalues, applicable to Sturm-Liouville problems.

Area of Science:

  • Differential Equations
  • Mathematical Physics
  • Spectral Theory

Background:

  • Second-order linear ordinary differential equations with specific potential functions exhibit discrete eigenvalues.
  • These eigenvalues are associated with a complete set of eigenfunctions.
  • Understanding eigenvalue behavior is crucial in various physics and engineering applications.

Purpose of the Study:

  • To derive a novel inequality involving the discrete eigenvalues {lambda(n)} of the given differential equation.
  • To establish explicit lower bounds for these eigenvalues using an elementary method.
  • To demonstrate the applicability of the method to classical Sturm-Liouville problems.

Main Methods:

  • An elementary and quick method was employed, avoiding complex comparison theorems.

Related Experiment Videos

  • The derivation focuses on establishing a direct relationship for the eigenvalues.
  • The method is general and does not require special assumptions on the potential function q(x).

    • Main Results:

    • A new inequality containing the discrete eigenvalues lambda(n) was successfully derived.
    • Explicit lower bounds for lambda(n) were obtained and demonstrated with three examples.
    • The method proved effective in providing respectable lower bounds for the classical Sturm-Liouville case.

    Conclusions:

    • The developed elementary method offers an efficient way to bound eigenvalues of specific differential equations.
    • The derived inequality and bounds are valuable for analyzing spectral properties.
    • The approach is versatile and extends to broader classes of differential equations.