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.
Cite this article
Jussi Behrndt, Roland Möws, Carsten Trunk, Eigenvalue estimates for singular left-definite Sturm–Liouville operators. J. Spectr. Theory 1 (2011), no. 3, pp. 327–347DOI 10.4171/JST/14