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;ℏ) of the problem−ℏ2ψ″(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;ℏ)=Sij(0)(E;ℏ)(1+ℏσij(E;ℏ)) where Sij(0)(E;ℏ) is the WKB approximation relative to the modified potential V(x)+ℏ24〈x〉−2 and the correction terms σij satisfy |∂Ekσij(E;ℏ)|⩽CkE−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.