# Eigenvalue estimates for singular left-definite Sturm–Liouville operators

### Jussi Behrndt

TU Graz, Austria### Roland Möws

TU Ilmenau, Germany### Carsten Trunk

TU Ilmenau, Germany

## 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.

## 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. 327–347

DOI 10.4171/JST/14