Singular Sturm–Liouville problems whose coefficients depend rationally on the eigenvalue parameter

Abstract

Let −Dω(·, z)D + q be a differential operator in L 2 (0, ∞) whose leading coefficient contains the eigenvalue parameter z. For the case that ω(·, z) has the particular formand the coefficient functions satisfy certain local integrability conditions, it is shown that there is an analog for the usual limit-point/limit-circle classification. In the limit-point case mild sufficient conditions are given so that all but one of the Titchmarsh-Weyl coefficients belong to the so-called Kac subclass of Nevanlinna functions. An interpretation of the Titchmarsh-Weyl coefficients is given also in terms of an associated system of differential equations where the eigenvalue parameter appears linearly.  2004 Elsevier Inc. All rights reserved.

Essential spectra of singular matrix differential operators of mixed order in the limit circle case

“…The formally symmetric operator T is said to be in the limit point case at the end point a (or b ) if exactly one linearly independent solution of ( T − λ) y = 0 is square integrable on ( a , c ] (or on [ c , b )) for a c ∈ ( a , b ) and T is said to be in the limit circle case at the endpoint a (or b ) if all solutions of ( T − λ) y = 0 are square integrable on ( a , c ] (or on [ c , b )), where \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\lambda \in {\mathbb C}\setminus {\mathbb R} $\end{document}. Similar results have been obtained by Hassi, Möller and de Snoo for singular λ‐rational Sturm‐Liouville equations in 6.…”

A class of Nevanlinna functions related to singular Sturm-Liouville problems

“…The exceptional value τ = −1/γ in Proposition 5.1 corresponds to the generalized Friedrichs extension; this extension coincides with the usual Friedrichs extension if the underlying symmetric operator is semibounded; cf. [6], [7], [8], [9], [10].…”

“…The Kac class N 1 is the collection of all Nevanlinna functions Q(z) for which For more on these classes, see [11], [12], [7,Section 1]. It has been shown recently (see [8, In fact, the arguments in [8], [9], [10] show that the corresponding TitchmarshWeyl coefficients have additional properties. To describe these properties, the class M 1 is introduced, which consists of all functions Q(z) ∈ N for which there exists a number c ∈ R such that…”

Limit‐point/limit‐circle classification for Hain–Lüst type equations

“…The first part of the Weyl approach, the pointwise limit‐point/limit‐circle alternative for , is based on Green's formula for , and as such is a generalization of the results obtained in for the case . Subject to a suitable definition of the quasi‐derivative, the proof is identical to that in the classical case.…”

“…Although the differential equation (1.2) for 1 can be considered in its own right, 2 given by (1.4) is needed to find the appropriate regularity condition for the solutions, i. e., ∈ ( 2 [0, ∞) ) 2 rather than 1 ∈ 2 [0, ∞) turns out to be the adequate criterion for the limit-point/limit-circle classification. The first part of the Weyl approach, the pointwise limit-point/limit-circle alternative for ∈ ℂ ⧵ ℝ, is based on Green's formula for , and as such is a generalization of the results obtained in [13] for the case = 0. Subject to a suitable definition of the quasi-derivative, the proof is identical to that in the classical case.…”