Power asymptotics of spectral functions of boundary value problems for generalized second-order differential equations with boundary conditions at a singular endpoint

Abstract

Let I = (-∞, b), where b ⩽ +∞, and let M(x), x ∈ I, be a nondecreasing function on I such that M(x) > 0 for x ∈ I. In the middle of the past century, it was proved that, in the case where M(x) is Lebesgue integrable on the interval (-∞, c), c ∈ I, the boundary value problem \(- \tfrac{d} {{dM(x)}}y^ + (x) = \lambda y(x)\), x ∈ I, limx→−∞y(x) = 1 is uniquely solvable for any complex λ and has at least one spectral function τ(λ) (“+” denotes right derivative).