Abstract view
SturmLiouville Problems Whose Leading Coefficient Function Changes Sign


Published:20030801
Printed: Aug 2003
Xifang Cao
Qingkai Kong
Hongyou Wu
Anton Zettl
Abstract
For a given SturmLiouville equation whose leading coefficient
function changes sign, we establish inequalities among the eigenvalues
for any coupled selfadjoint boundary condition and those for two
corresponding separated selfadjoint boundary conditions. By a recent
result of Binding and Volkmer, the eigenvalues (unbounded from both
below and above) for a separated selfadjoint 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 selfadjoint
boundary condition. Under this indexing scheme, we determine the
discontinuities of each eigenvalue as a function on the space of such
SturmLiouville problems, and its range as a function on the space of
selfadjoint 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
SturmLiouville 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  SturmLiouville 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
