On a necessary aspect for the Riesz basis property for indefinite Sturm-Liouville problems

Abstract

In 1996, H. Volkmer observed that the inequality \[(\int_{-1}^1\frac{1}{|r|}|f'|dx)^2 \le K^2 \int_{-1}^1|f|^2dx\int_{-1}^1\Big|\Big(\frac{1}{r}f'\Big)'\Big|^2dx \] is satisfied with some positive constant $K>0$ for a certain class of functions $f$ on $[-1,1]$ if the eigenfunctions of the problem \[ -y"=λ\, r(x)y,\quad y(-1)=y(1)=0 \] form a Riesz basis of the Hilbert space $L_{|r|}^2(-1,1)$. Here the weight $r\in L^1(-1,1)$ is assumed to satisfy $xr(x)>0$ a.e. on $[-1,1]$. We present two criteria in terms of Weyl-Titchmarsh $m$-functions for the Volkmer inequality to be valid. Using these results we show that this inequality is valid if the operator associated with the spectral problem satisfies the linear resolvent growth condition. In particular, we show that the Riesz basis property of eigenfunctions is equivalent to the linear resolvent growth if $r$ is odd.

26 pages

References in corpus (4)

• The similarity problem for $J$-nonnegative Sturm-Liouville operators
• The similarity problem for indefinite Sturm-Liouville operators and the HELP inequality
• Indefinite Sturm-Liouville operators with the singular critical point zero
• The Riesz basis property of an indefinite Sturm-Liouville problem with non-separated boundary conditions

Cited by in corpus (1)

• The similarity problem for indefinite Sturm-Liouville operators and the HELP inequality