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

TL;DR

This paper explores conditions under which the eigenfunctions of an indefinite Sturm-Liouville problem form a Riesz basis, providing criteria based on Weyl-Titchmarsh functions and linking the basis property to resolvent growth, especially for odd weights.

Contribution

The paper introduces two new criteria involving Weyl-Titchmarsh m-functions for the Riesz basis property in indefinite Sturm-Liouville problems and connects this property to the linear resolvent growth condition.

Findings

Criteria for Volkmer inequality validity using Weyl-Titchmarsh functions.

Equivalence of Riesz basis property and linear resolvent growth for odd weights.

Validation of inequality under spectral operator conditions.

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 \[ -y"=\lambda\, r(x)y,\quad y(-1)=y(1)=0 \] form a Riesz basis of the Hilbert space $L^2_{|r|}(-1,1)$. 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.