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Connection formulae for spectral functions associated with singular Sturm–Liouville equations

Published online by Cambridge University Press:  11 July 2007

D. J. Gilbert
Affiliation:
Sheffield Hallam University, City Campus, Howard Street, Sheffield S1 1WB, UK
B. J. Harris
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, Dekalb, IL 60115, USA

Abstract

We consider the Sturm–Liouville equation with the initial condition and suppose that Weyl's limit-point case holds at infinity. Let ρα(μ) be the corresponding spectral function and its symmetric derivative. We show that for almost all μ ∈ R, if exists and is positive for some α ∈ [0, π), then (i) exists and is positive for all β ∈ [0, π), and (ii) for all α1, α2 ∈ (0, π) \ {½ π},

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2000

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