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

Abstract

This paper studies the scattering matrix $\Sigma (E;\hslash )$ of the problem \[ -\hbar^2 ψ''(x) + V(x) ψ(x) = Eψ(x) \] for positive potentials $V\in C^{\infty }(\mathbb{R})$ with inverse square behavior as $x\to \pm \infty$. It is shown that each entry takes the form ${\Sigma }_{ij}(E;\hslash )={\Sigma }_{ij}^{(0)}(E;\hslash )(1+\hslash {\sigma }_{ij}(E;\hslash ))$ where ${\Sigma }_{ij}^{(0)}(E;\hslash )$ 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 $|{\partial }_E^k{\sigma }_{ij}(E;\hslash )|\le C_k{E}^{-k}$ for all $k\ge 0$ and uniformly in $(E,\hslash )\in (0,E_0)\times (0,{\hslash }_0)$ where $E_0,{\hslash }_0$ are small constants. This asymptotic behavior is not universal: if $-{\hslash }^2{\partial }_x^2+V$ has a {\em zero energy resonance}, then $\Sigma (E;\hslash )$ exhibits different asymptotic behavior as $E\to 0$. The resonant case is excluded here due to $V>0$.

References in corpus (2)

• Dispersive estimates for Schrodinger operators in dimensions one and three
• Quantum scattering at low energies