We prove that, for a density of disorder $ρ$ small enough, a certain class of discrete random Schrödinger operators on $\Z^d$ with diluted potentials exhibits a Lifschitz behaviour from the bottom of the spectrum up to energies at a distance of the order $ρ^α$ from the bottom of the spectrum, with $α>2(d+1)/d$. This leads to localization for the energies in this zone for these low density models. The same results hold for operators on the continuous, and in particular, with Bernoulli or Poisson random potential.
We prove exponential decay for a system of two Schr{ö}dinger equations in a wave guide, with coupling and damping at the boundary. This relies on the spectral analysis of the corresponding coupled Schr{ö}dinger operator on the one-dimensional cross section. We show in particular that we have a spectral gap and that the corresponding generalized eigenfunctions form a Riesz basis.
We prove low frequency resolvent estimates and local energy decay for the Schr{ö}dinger equation in an asymptotically Euclidean setting. More precisely, we go beyond the optimal estimates by comparing the resolvent of the perturbed Schr{ö}dinger operator with the resolvent of the free Laplacian. This gives the leading term for the developpement of this resolvent when the spectral parameter is close to 0. For this, we show in particular how we can apply the usual commutators method for generalized resolvents and simultaneously for different operators. Finally, we deduce similar results for the …
We consider discrete Schr{ö}dinger operators on ${\mathbb{Z}}^d$ for which the perturbation consists of the sum of a long-range type potential and a Wigner-von Neumann type potential. Still working in a framework of weighted Mourre theory, we improve the limiting absorption principle (LAP) that was obtained in [Ma1]. To our knowledge, this is a new result even in the one-dimensional case. The improvement consists in a weakening of the assumptions on the long-range potential and better LAP weights. The improvement relies only on the fact that the generator of dilations (which serves as conjugat…
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。
How and why may an interacting system of many particles be described assuming that all particles are independent and identically distributed ? This question is at least as old as statistical mechanics itself. Its quantum version has been rejuvenated by the birth of cold atoms physics. In particular the experimental creation of Bose-Einstein condensates directly asks the following variant: why and how can a large assembly of very cold interacting bosons (quantum particles deprived of the Pauli exclusion principle) all populate the same quantum state ? In this text I review the various mathemati…
