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Eigenvalue estimates for singular left-definite Sturm-Liouville operators
The spectral properties of a singular left-definite Sturm-Liouville operator
are investigated and described via the properties of the corresponding
right-definite selfadjoint counterpart which is obtained by substituting
the indefinite weight function by its absolute value. The spectrum of the
-selfadjoint operator is real and it follows that an interval
is a gap in the essential spectrum of if and only
if both intervals and are gaps in the essential spectrum of
the -selfadjoint operator . As one of the main results it is shown that
the number of eigenvalues of in differs at most by
three of the number of eigenvalues of in the gap ; as a byproduct
results on the accumulation of eigenvalues of singular left-definite
Sturm-Liouville operators are obtained. Furthermore, left-definite problems
with symmetric and periodic coefficients are treated, and several examples are
included to illustrate the general results.Comment: to appear in J. Spectral Theor