Eigenvalue estimates for singular left-definite Sturm–Liouville operators

Abstract

The spectral properties of a singular left-definite Sturm–Liouville operator JA are investigated and described via the properties of the corresponding right-definite selfadjoint counterpart A which is obtained by substituting the indefinite weight function by its absolute value. The spectrum of the J -selfadjoint operator JA is real and it follows that an interval (a,b)\subset\mathbb R^+ is a gap in the essential spectrum of A if and only if both intervals (-b,-a) and (a,b) are gaps in the essential spectrum of the J -selfadjoint operator JA . As one of the main results it is shown that the number of eigenvalues of JA in (-b,-a) \cup (a,b) differs at most by three of the number of eigenvalues of A in the gap (a,b) ; 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.