JournalsjstVol. 1, No. 3pp. 327–347

Eigenvalue estimates for singular left-definite Sturm–Liouville operators

  • Jussi Behrndt

    TU Graz, Austria
  • Roland Möws

    TU Ilmenau, Germany
  • Carsten Trunk

    TU Ilmenau, Germany
Eigenvalue estimates for singular left-definite Sturm–Liouville operators cover
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Abstract

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