Analytic continuation and resonance-free regions for Sturm-Liouville potentials with power decay

Let us write λ = z, where 0 ≤ arg z < π when 0 ≤ arg λ < 2π. Then (1. 3) implies that there is a solution ψ(x, z) of (1. 1) such that ψ(x, z) ∼ exp(izx), ψ′(x, z) ∼ iz exp(izx) (1. 4) as x → ∞, and ψ(x, z) is analytic in z for im z > 0 [7, Theorem 1.9.1]. Then ψ(x, z) is the Weyl L2(0,∞) solution of (1. 1) when λ is non-real and it forms the basis of the Weyl-Titchmarsh spectral theory of (1. 1) [5, Chapter 9], [17],[18]. A central result of this spectral theory is the existence of a spectral function ρα(μ) (−∞ < μ < ∞) which is piecewise constant in (−∞, 0) and locally absolutely continuous in [0,∞) with ρ′α(μ) > 0 [17, section 5.7], [18, p. 264]. In particular, (1. 4) leads to the Kodaira formula