Semiclassical analysis of low and zero energy scattering for one-dimensional Schrödinger operators with inverse square potentials

Abstract

This paper studies the scattering matrix S (E ; ℏ) of the problem- ℏ2 ψ″ (x) + V (x) ψ (x) = E ψ (x) for positive potentials V ∈ C∞ (R) with inverse square behavior as x → ± ∞. It is shown that each entry takes the form Si j (E ; ℏ) = Si j(0) (E ; ℏ) (1 + ℏ σi j (E ; ℏ)) where Si j(0) (E ; ℏ) is the WKB approximation relative to the modified potentialV (x) + frac(ℏ2, 4) 〈 x 〉-2 and the correction terms σi j satisfy | ∂Ek σi j (E ; ℏ) | ≤ Ck E- k for all k ≥ 0 and uniformly in (E, ℏ) ∈ (0, E0) × (0, ℏ0) where E0, ℏ0 are small constants. This asymptotic behavior is not universal: if - ℏ2 ∂x2 + V has a zero energy resonance, then S (E ; ℏ) exhibits different asymptotic behavior as E → 0. The resonant case is excluded here due to V > 0. © 2008 Elsevier Inc. All rights reserved.