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

Abstract

In 1996, H. Volkmer observed that the inequality ()∫−111|r||f′|2dx2≤K2∫−11|f|2dx∫−11||()1rf′′2dxis satisfied with some positive constant K>0 for a certain class of functions f on [ − 1, 1] if the eigenfunctions of the problem −y′′=λr(x)y,y(−1)=y(1)=0form a Riesz basis of the Hilbert space L|r|2(−1,1). Here the weight r∈L1(−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. Note that one of these criteria is new even for the classical HELP inequality. Using these results we improve the result of Volkmer by showing 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.

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

“…We complete this section with the following Remark 4.21. In the regular case, Theorem 4.20 was established in [50]. Moreover, in this case the implication (i) ⇒ (iv) was observed by Volkmer [65] and the converse implication (iv) ⇒ (i) was noticed in [10].…”

“…is necessary for the similarity of a closed linear operator T acting in a Hilbert space H to a self-adjoint operator. It was noticed in [50,Theorem 7.3] that in the regular case, i.e., b < ∞ and q, w, r ∈ L 1 (−b, b) are even, condition (4.33) is necessary and sufficient for the operator A to be similar to a self-adjoint operator. Moreover, in [65], the connection between the similarity problem and the HELP inequality was observed.…”

“…In the regular case, this criterion was established in [50] and in contrast to the classical Everitt criterion, this result shows that it suffices to know the behavior of m(.) only along the imaginary semi-axis.…”

“…Motivated by various problems arising in physics, scattering and transport theory [5,7,26,30,39,63], the problem of whether or not the eigenfunctions of (1.1) form a Riesz basis of L 2 w (−b, b) attracted a lot of attention since the mid of seventies of the last century (see e.g. [4,6,10,11,17,18,19,25,27,28,29,40,41,42,43,48,49,50,56,57,58,65] and references therein). The first general sufficient condition for the Riesz basis property was obtained by Beals in [6] and later this condition has been extended and generalized by many authors (for a survey we refer to the recent papers [11,17], see also [10,18,27]).…”

Spectral analysis of indefinite Sturm-Liouville operators

“…It is known [6] that the spectrum of (2) is discrete and symmetric with respect to R, and the nonreal part of the spectrum consists of finitely many eigenvalues of finite algebraic and geometric multiplicity. There is a vast literature devoted to the study of the Riesz basis property in the Hilbert space L 2 w (−1, 1) of a system of eigen-and associated functions of (2) (see [2], [4]- [6], [11], [14], [15], and the references therein). The existence of weights w such that the eigen-and associated functions of (2) do not form a Riesz basis was proved by H. Volkmer, and the first explicit examples were constructed later by A. Fleige, N. L. Abasheeva, and S. G. Pyatkov (see [4] and [14]).…”