Semiclassical analysis of low and zero energy scattering for one dimensional Schrödinger operators with inverse square potentials
| dc.creator | Costin, Ovidiu | |
| dc.creator | Schlag, Wilhelm | |
| dc.creator | Staubach, Wolfgang | |
| dc.creator | Tanveer, Saleh | |
| dc.date | 2008-04-14 | |
| dc.date | 2008-04-16 | |
| dc.date.accessioned | 2026-07-07T09:32:39Z | |
| dc.date.available | 2026-07-07T09:32:39Z | |
| dc.description | This paper studies the scattering matrix $Σ(E;\hbar)$ of the problem \[ -\hbar^2 ψ''(x) + V(x) ψ(x) = Eψ(x) \] 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 $Σ_{ij}(E;\hbar)=Σ_{ij}^{(0)}(E;\hbar)(1+\hbar σ_{ij}(E;\hbar))$ where $Σ_{ij}^{(0)}(E;\hbar)$ is the WKB approximation relative to the {\em modified potential} $V(x)+\frac{\hbar^2}{4} \la x\ra^{-2}$ and the correction terms $σ_{ij}$ satisfy $|\partial_E^k σ_{ij}(E;\hbar)| \le C_k E^{-k}$ 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 $Σ(E;\hbar)$ exhibits different asymptotic behavior as $E\to0$. The resonant case is excluded here due to $V>0$. | |
| dc.identifier | https://arxiv.org/abs/0804.2282 | |
| dc.identifier | http://arxiv.org/abs/0804.2282 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/158854 | |
| dc.subject | Mathematical Physics | |
| dc.subject | 81Q20;34L25;34B25 | |
| dc.title | Semiclassical analysis of low and zero energy scattering for one dimensional Schrödinger operators with inverse square potentials | |
| dc.type | text |