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

This paper studies the scattering matrix S(E; ¯h) of the problem −¯h2ψ (x)+V (x)ψ(x) = Eψ(x) for positive potentials V ∈ C∞(R) with inverse square behavior as x →±∞. It is shown that each entry takes the form Sij (E; ¯h) = S(0) ij (E; ¯h)(1 + ¯hσij (E; ¯h)) where S(0) ij (E; ¯h) is the WKB approximation relative to the modified potential V (x)+ ¯h2 4 x −2 and the correction terms σij satisfy |∂k Eσij (E; ¯h)| CkE−k for all k 0 and uniformly in (E, ¯h) ∈ (0,E0) × (0, ¯h0) where E0, ¯h0 are small constants. This asymptotic behavior is not universal: if −¯h2∂2 x + V has a zero energy resonance, then S(E; ¯h) exhibits different asymptotic behavior as E→0. The resonant case is excluded here due toV >0. © 2008 Elsevier Inc. All rights reserved.