Sturm-Liouville Problems Whose Leading Coefficient Function Changes Sign

[摘要] For a given Sturm-Liouville equation whose leading coefficientfunction changes sign, we establish inequalities among the eigenvaluesfor any coupled self-adjoint boundary condition and those for twocorresponding separated self-adjoint boundary conditions. By a recentresult of Binding and Volkmer, the eigenvalues (unbounded from bothbelow and above) for a separated self-adjoint boundary condition canbe numbered in terms of the Pr"ufer angle; and our inequalities canthen be used to index the eigenvalues for any coupled self-adjointboundary condition. Under this indexing scheme, we determine thediscontinuities of each eigenvalue as a function on the space of suchSturm-Liouville problems, and its range as a function on the space ofself-adjoint boundary conditions. We also relate this indexing schemeto the number of zeros of eigenfunctions. In addition, wecharacterize the discontinuities of each eigenvalue under a differentindexing scheme.

[学科分类] 数学（综合）