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Sturm-Liouville Problems Whose Leading Coefficient Function Changes Sign

  Published:2003-08-01
 Printed: Aug 2003
  • Xifang Cao
  • Qingkai Kong
  • Hongyou Wu
  • Anton Zettl
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Abstract

For a given Sturm-Liouville equation whose leading coefficient function changes sign, we establish inequalities among the eigenvalues for any coupled self-adjoint boundary condition and those for two corresponding separated self-adjoint boundary conditions. By a recent result of Binding and Volkmer, the eigenvalues (unbounded from both below and above) for a separated self-adjoint boundary condition can be numbered in terms of the Pr\"ufer angle; and our inequalities can then be used to index the eigenvalues for any coupled self-adjoint boundary condition. Under this indexing scheme, we determine the discontinuities of each eigenvalue as a function on the space of such Sturm-Liouville problems, and its range as a function on the space of self-adjoint boundary conditions. We also relate this indexing scheme to the number of zeros of eigenfunctions. In addition, we characterize the discontinuities of each eigenvalue under a different indexing scheme.
MSC Classifications: 34B24, 34C10, 34L05, 34L15, 34L20 show english descriptions Sturm-Liouville theory [See also 34Lxx]
Oscillation theory, zeros, disconjugacy and comparison theory
General spectral theory
Eigenvalues, estimation of eigenvalues, upper and lower bounds
Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions
34B24 - Sturm-Liouville theory [See also 34Lxx]
34C10 - Oscillation theory, zeros, disconjugacy and comparison theory
34L05 - General spectral theory
34L15 - Eigenvalues, estimation of eigenvalues, upper and lower bounds
34L20 - Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions
 

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