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 ; ?) of the problem- ?² ?^? (x) + V (x) ? (x) = E ? (x) for positive potentials V ? C⁸ (R) with inverse square behavior as x ? ± 8. It is shown that each entry takes the form S_{i j} (E ; ?) = S_{i j}⁽⁰⁾ (E ; ?) (1 + ? s_{i j} (E ; ?)) where S_{i j}⁽⁰⁾ (E ; ?) is the WKB approximation relative to the modified potentialV (x) + frac(?², 4) <x >^-2 and the correction terms s_{i j} satisfy | ?_E^k s_{i j} (E ; ?) | = C_k E^{- k} for all k = 0 and uniformly in (E, ?) ? (0, E₀) × (0, ?₀) where E₀, ?₀ are small constants. This asymptotic behavior is not universal: if - ?² ?_x² + V has a zero energy resonance, then S (E ; ?) exhibits different asymptotic behavior as E ? 0. The resonant case is excluded here due to V > 0. © 2008 Elsevier Inc. All rights reserved.