Scattering theory for the Schrödinger equation with repulsive potential
Scattering theory for the Schrödinger equation with repulsive potential
复制标题
具有排斥势的薛定谔方程的散射理论
DOI：
10.1016/j.matpur.2004.10.007
复制
发表时间：
2004
期刊：
影响因子：
--
通讯作者：
L. Michel
中科院分区：
文献类型：
--
作者：
J. Bony;R. Carles;D. Haefner;L. Michel
我们考虑以−Δ−|x|α为参考哈密顿量的薛定谔方程的散射理论，其中0和lt；α⩽2在任意空间维。我们证明，当这个哈密顿量被一个势扰动时，通常的短程/长程条件被削弱：势的极限衰变取决于α的值，并且在未扰动的情况下与经典轨迹的增长有关。借助于新的共轭算子的Mourre估计，建立了波算子的存在性及其渐近完备性。我们构造了渐近速度，并描述了它的谱。将一些结果推广到用一般二阶多项式代替−|x|α的情形。
We consider the scattering theory for the Schrödinger equation with −Δ−|x|αas a reference Hamiltonian, for 0<α⩽2, in any space dimension. We prove that, when this Hamiltonian is perturbed by a potential, the usual short range/long range condition is weakened: the limiting decay for the potential depends on the value of α, and is related to the growth of classical trajectories in the unperturbed case. The existence of wave operators and their asymptotic completeness are established thanks to Mourre estimates relying on new conjugate operators. We construct the asymptotic velocity and describe its spectrum. Some results are generalized to the case where −|x|αis replaced by a general second order polynomial.