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

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

2008-04-14

arxiv.org/abs/0804.2282v2

Physics

Mathematical Physics

Scientific paper

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 $V>0$.

Affiliated with

Also associated with

No associations

Rating

Semiclassical analysis of low and zero energy scattering for one dimensional Schrödinger operators with inverse square potentials does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Semiclassical analysis of low and zero energy scattering for one dimensional Schrödinger operators with inverse square potentials, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Semiclassical analysis of low and zero energy scattering for one dimensional Schrödinger operators with inverse square potentials will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-652436