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

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},