Compactness conditions for the green operator in Lp(double-struck R sign) corresponding to a general Sturm-Liouville operator

N. Chernyavskaya, L. Shuster

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

We consider the equation (0.1) -(r(x)y′(x))′ + q(x)y(x) = f(x), x ∈ double-struck R sign, where r(x) > 0, q(x) ≥ 0 for x ∈ double-struck R sign, 1/r(x) ∈ Lloc1(double-struck R sign), q(x) ∈ Lloc1 (double-struck R sign), f(x) ∈ Lp(double-struck R sign), p ∈ [1, ∞] (L(double-struck R sign) := C(double-struck R sign)). We give necessary and sufficient conditions under which, regardless of p ∈ [1, ∞], the following statements hold simultaneously: I) For any f(x) ∈ Lp(double-struck R sign) Equation (0.1) has a unique solution y(x) ∈ Lp(double-struck R sign) where y(x) = (Gf)(x) def= ∫-∞ G(x,t)f(t)dt, x ∈ double-struck R sign. II) The operator G : Lp (double-struck R sign) → Lp(double-struck R sign) is compact. Here G(x,t) is the Green function corresponding to (0.1). This result is applied to study some properties of the spectrum of the Sturm-Liouville operator.

Original languageEnglish
Pages (from-to)33-53
Number of pages21
JournalMathematische Nachrichten
Volume215
DOIs
StatePublished - 2000

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