The Aharonov-Bohm effect is a fundamental and controversial issue in physics. At stake are what are the fundamental electromagnetic quantities in quantum physics, if magnetic fields can act at a distance on charged particles and if the magnetic potentials have a real physical significance. From the experimental side the issues were settled by the remarkable experiments of Tonomura et al. in 1982 and 1986 with toroidal magnets that gave a strong experimental evidence of the physical existence of the Aharonov-Bohm effect, and by the recent experiment of Caprez et al. in 2007 that shows that the results of these experiments can not be explained by a force. The Aharonov-Bohm Ansatz of 1959 predicts the results of the experiments of Tonomura et al. and of Caprez et al. In 2009 we gave the first rigorous proof that the Aharonov-Bohm Ansatz is a good approximation to the exact solution for toroidal magnets under the conditions of the experiments of Tonomura et al.. In this paper we prove that our results do not depend on the particular geometry of the magnets and on the velocities of the incoming electrons used on the experiments, and on the gaussian shape of the wave packets used to obtain our quantitative error bound. We consider a general class of magnets that are a finite union of handle bodies. We formulate the Aharonov-bohm Ansatz that is appropriate to this general case and we prove that the exact solution to the Schroedinger equation is given by the Aharonov-Bohm Ansatz up to an error bound in norm that is uniform in time and that decays as a constant divided by $v^ρ, 0 < ρ<1$, with $v$ the velocity. The results of Tonomura et al., of Caprez et al., our previous results and the results of this paper give a firm experimental and theoretical basis to the existence of the Aharonov-Bohm effect and to its quantum nature.
On the ground state of the nonlinear Schr{ö}dinger equation: asymptotic behavior at the endpoint powers
CNRS IRMAR - UMR 6625（法国国家科学研究中心 IRMAR）；Univ. Lille, CNRS, Inria UMR 8524 - Laboratoire Paul Painlevé（里尔大学，法国国家科学研究中心，Inria UMR 8524 - 保罗·庞加莱实验室）；Department of Mathematics, McMaster University（麦克马斯特大学数学系）
On unique continuation for solutions of the Schr{ö}dinger equation on trees
我们证明了如果齐次树上的含时薛定谔方程的解在两个不同的时刻快速衰减，则该解是平凡的.对于自由薛定谔算子，我们利用Laplacian谱理论和复分析，得到了导致在任意时刻急剧衰减的初始条件的一个特征.然后，我们利用Colin de Verdi和Turc的紧支势的Schr dinger算子的谱分解，推广了我们在紧支势存在下的结果.最后，我们使用实变量的方法首先介绍了Escauriaza，Kenig，Ponce和Vega建立一个一般的尖锐的结果的情况下，有界的潜力。
