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

This paper studies the scattering matrix S(E; h) of the problem -h(2)psi"(x) + V(x)psi(x) = E psi(x) for positive potentials V is an element of C-infinity(R) with inverse square behavior as x -> +/-infinity. It is shown that each entry takes the form S-ij (E; h) = S-ij((0))(E; h)(1 + h sigma(ij) (E; h)) where S-ij((0))(E; h) is the WKB approximation relatj tive to the modifiedpotential V(x) + n(2)/4 < x >(-2) and the correction terms sigma(ij) satisfy |partial derivative(k)(E)sigma(ij)(E; h| <= CkE-k for all k >= 0 and uniformly in (E, h) is an element of (0, E-0) x (0, h(0)) where E-0, h(0) are small constants. This asymptotic behavior is not universal: if -h(2)partial derivative(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 to V > 0. (C) 2008 Elsevier Inc. All rights reserved.