Semiclassical Analysis of Low and Zero Energy Scattering for One Dimensional Schrödinger Operators with Inverse Square Potentials
نویسنده
چکیده
This paper studies the scattering matrix S(E;~) of the problem −~ψ(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;~) = S (0) ij (E;~)(1 + ~σij(E;~)) where S (0) ij (E;~) is the WKB approximation relative to the modified potential V (x) + ~ 2 4 〈x〉−2 and the correction terms σij satisfy |∂ k Eσij (E;~)| ≤ CkE −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∂2 x + 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.
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