Estimates on the non-real eigenvalues of regular indefinite Sturm–Liouville problems

Abstract

Abstract. Regular Sturm-Liouville problems with indefinite weight functions may possess finitely many non-real eigenvalues. In this note we prove explicit bounds on the real and imaginary parts of these eigenvalues in terms of the coefficients of the differential expression.

Spectral bounds for indefinite singular Sturm–Liouville operators with uniformly locally integrable potentials

“…The main objective of the present paper is to prove bounds on the absolute values and the imaginary parts of the non-real eigenvalues of A. For the case of regular indefinite Sturm-Liouville operators related estimates were obtained in [1,8,9,14,17,20]. The more difficult case of singular indefinite Sturm-Liouville operators was so far only studied in very special situations; cf.…”

Spectral bounds for singular indefinite Sturm-Liouville operators with 𝐿¹-potentials

“…In this situation the spectrum of A is purely discrete and various estimates on the real and imaginary parts of the non-real eigenvalues were obtained in the last few years; cf. [2,9,10,14,17,21]. The singular case is much less studied, due to the technical difficulties which, very roughly speaking, are caused by the presence of continuous spectrum.…”

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.…”

“…Regarding the regular case, i.e. the Sturm-Liouville differential expression is defined on a finite interval with integrable coefficients, bounds in terms of the coefficients can be found in [2,7,10]; we also mention that the techniques in [1, Section 3] may be used to prove related eigenvalue estimates.…”