Non-real eigenvalues of singular indefinite Sturm-Liouville operators

Abstract

Abstract. We study a Sturm-Liouville expression with indefinite weight of the form sgn(−d 2 /dx 2 + V ) on R and the non-real eigenvalues of an associated selfadjoint operator in a Krein space. For real-valued potentials V with a certain behaviour at ±∞ we prove that there are no real eigenvalues and that the number of non-real eigenvalues (counting multiplicities) coincides with the number of negative eigenvalues of the selfadjoint operator associated toThe general results are illustrated with examples.

Bounds on the Non‐real Spectrum of a Singular Indefinite Sturm‐Liouville Operator on ℝ

“…The spectral properties of such differential operators have attracted interest for more than a century, see [8,11]. For an overview we refer to [13] and for recent results on the non-real spectrum see [2][3][4][5][6][7]10]. The main objective of this note is to proof an estimate on the absolute values of the non-real eigenvalues of the indefinite Sturm-Liouville operator A in (1) which depends only on the L 1 -norm of the continuous potential q.…”

Locally Definitizable Operators: The Local Structure of the Spectrum

“…[15,21,22,25,31,34,38,39,41,66,93], KreinFeller operators [46], λ-dependent boundary value problems, see e.g. [24,30,42,62,63,82], operator polynomials [68,69,71,72,74,75,76,87], second order systems [50,89,90] and in the study of problems of Klein-Gordon type [83].…”

Eigenvalues of Schrödinger operators with definite and indefinite weights

“…Many authors in applied and pure mathematics used Titchmarsh-Weyl's theory to compute the eigenvalues of Schrödinger operators, Sturm-Liouville operators with definite and indefinite weights [5,7,8,23]. This paper develops a methodology to compute the eigenvalues for Sturm-Liouville problems of the following forms, ðHyÞðxÞ ¼ Ày 00 ðxÞ þ VðxÞyðxÞ ð Þ ¼ kyðxÞ; x 2 R; y 2 L 2 ðRÞ; ð1Þ…”

“…Recently, in [7], the eigenvalues of Cut-Off Coulomb potential and the Sech-Squared problem [8] have been computed with indefinite weight and we have been able to analyze the behavior of the spectrum. The computations of nonreal eigenvalues of these problems consequently led to the proof of many theoretical results in this area of indefinite problems.…”