Semiclassical analysis of low and zero energy scattering for one dimensional Schr\"odinger operators with inverse square potentials

TL;DR

This paper provides a semiclassical analysis of the scattering matrix for one-dimensional Schrödinger operators with inverse square potentials, detailing the asymptotic behavior near zero energy and the effects of zero energy resonances.

Contribution

It introduces a refined asymptotic description of the scattering matrix entries using WKB approximation with a modified potential, including correction bounds, for positive inverse square potentials.

Findings

Scattering matrix entries approximate WKB solutions with correction terms.

Asymptotic behavior differs in the presence of zero energy resonances.

Correction terms are uniformly bounded in energy and Planck's constant.

Abstract

This paper studies the scattering matrix $\Sigma(E;\hbar)$ of the problem \[ -\hbar^2 \psi''(x) + V(x) \psi(x) = E\psi(x) \] for positive potentials $V\in C^\infty(\R)$ with inverse square behavior as $x\to\pm\infty$. It is shown that each entry takes the form $\Sigma_{ij}(E;\hbar)=\Sigma_{ij}^{(0)}(E;\hbar)(1+\hbar \sigma_{ij}(E;\hbar))$ where $\Sigma_{ij}^{(0)}(E;\hbar)$ is the WKB approximation relative to the {\em modified potential} $V(x)+\frac{\hbar^2}{4} \la x\ra^{-2}$ and the correction terms $\sigma_{ij}$ satisfy $|\partial_E^k \sigma_{ij}(E;\hbar)| \le C_k E^{-k}$ for all $k\ge0$ and uniformly in $(E,\hbar)\in (0,E_0)\times (0,\hbar_0)$ where $E_0,\hbar_0$ are small constants. This asymptotic behavior is not universal: if $-\hbar^2\partial_x^2 + V$ has a {\em zero energy resonance}, then $\Sigma(E;\hbar)$ exhibits different asymptotic behavior as $E\to0$. The resonant case is excluded here due to $V>0$.