Avy Soffer, Wei-Min Wang
We prove that at large disorder, Anderson localization in $\Z^d$ is stable under localized time-periodic perturbations by proving that the associated quasi-energy operator has pure point spectrum. The formulation of this problem is motivated by questions of Anderson localization for non-linear Schrödinger equations.
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On the canonical $2$-sphere and for Schr{ö}dinger eigenfunctions, we obtain a simple geometric criterion on the potential under which we can improve, near a given point and for every $p\neq 6$, Sogge's estimates by a power of the eigenvalue. This criterion can be formulated in terms of the critical points of the Radon transform of the potential and it is independent of the choice of eigenfunctions.
In this survey paper, we report on recent works concerning exact observability (and, by duality, exact controllability) properties of subelliptic wave and Schr{ö}dinger-type equations. These results illustrate the slowdown of propagation in directions transverse to the horizontal distribution. The proofs combine sub-Riemannian geometry, semi-classical analysis, spectral theory and non-commutative harmonic analysis.
An analogue of Rellich's theorem is proved for discrete Laplacian on square lattice, and applied to show unique continuation property on certain domains as well as non-existence of embedded eigenvalues for discrete Schr{ö}dinger operators.
Uniform observation of semiclassical Schr{ö}dinger eigenfunctions on an interval
本文研究区间上单阱势与 Dirichlet 边界条件下的半经典 Schrödinger 特征函数，利用 Agmon 估计、Gronwall 论证及半经典测度，建立在半经典和高能极限下均一致且最优的 L^2 密度上下界。
We prove a Limiting Absorption Principle for Schr{ö}dinger operators in tubes about infinite curves embedded in the Euclidian space with different types of boundary conditions. The argument is based on the Mourre theory with conjugate operators different from the generator of dilations which is usually used in this case, and permits to prove a Limiting Absorption Principle for Schr{ö}dinger operators in singular waveguides.
We recall a Moure theory adapted to non self-adjoint operators and we apply this theory to Schr{ö}dinger operators with non real potentials, using different type of conjugate operators. We show that some conjugate operators permits to relax conditions on the derivatives of the potential that were required up to now.
We compute an explicit formula for the integrated density of states of the periodic Airy-Schr{ö}dinger operator on the real line. For this purpose, we study precisely the spectrum of the restriction of the periodic Airy-Schr{ö}dinger operator to a finite number of periods. We prove that all the eigenvalues of this restriction are in the spectral bands of the periodic Airy-Schr{ö}dinger operator and none of them are in its spectral gaps. We count exactly the number of eigenvalues in each of the spectral bands, this number being equal to the number of periods to which the periodic operator is re…
On a complete weighted Riemannian manifold $(M^n,g,μ)$ satisfying the doubling condition and the Poincar{é} inequalities, we characterize the class of function $V$ such that the Schr{ö}dinger operator $Δ-V$ maps the homogeneous Sobolev space $W_o^{1,2} (M)$ to its dual space. On Euclidean space, this result is due to Maz'ya and Verbitsky. In the proof of our result, we investigate the weighted $L^2$-boundedness of the Hodge projector.
We give necessary and sufficient conditions for the controllability of a Schr\''odinger equation involving the sub-Laplacian of a nilmanifold obtained by taking the quotient of a group of Heisenberg type by one of its discrete sub-groups.This class of nilpotent Lie groups is a major example of stratified Lie groups of step 2. The sub-Laplacian involved in these Schr\''odinger equations is subelliptic, and, contrarily to what happens for the usual elliptic Schr\''odinger equation for example on flat tori or on negatively curved manifolds, there exists a minimal time of controllability. The main…
We study to what extent Lieb--Thirring inequalities are extendable from self-adjoint to general (possibly non-self-adjoint) Jacobi and Schrödinger operators. Namely, we prove the conjecture of Hansmann and Katriel from [Complex Anal. Oper. Theory 5, No. 1 (2011), 197-218] and answer another open question raised therein. The results are obtained by means of asymptotic analysis of eigenvalues of discrete Schrödinger operators with rectangular barrier potential and complex coupling. Applying the ideas in the continuous setting, we also solve a similar open problem for one-dimensional Schrödinger …
Long-time dynamics for the magnetic Schr{ö}dinger equation on the torus
本文研究二维环面上磁薛定谔流的长时间半经典测度，在几何非消失条件下证明量子唯一遍历性的动力学形式，并给出构型与动量边际的收敛阈值。
Spectral flow and Levinson's theorem for Schrö dinger operators
本文利用谱流为具有光滑紧支撑势的 Schrödinger 算子提供了 Levinson 定理的新证明，该证明适用于所有维度及存在共振的情况，并基于谱位移函数与高能修正时间延迟表述结论。
We study an inverse scattering problem at fixed energy for radial magnetic Schr{ö}dinger operators on R^2 \ B(0, r\_0), where r\_0 is a positive and arbitrarily small radius. We assume that the magnetic potential A satisfies a gauge condition and we consider the class C of smooth, radial and compactly supported electric potentials and magnetic fields denoted by V and B respectively. If (V, B) and (\tilde{V} , \tilde{B}) are two couples belonging to C, we then show that if the corresponding phase shifts $δ$\_l and \tilde{$δ$}\_l (i.e. the scattering data at fixed energy) coincide for all l $\in…
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…
We prove Anderson localization at the internal band-edges for periodic magnetic Schr{ö}dinger operators perturbed by random vector potentials of Anderson-type. This is achieved by combining new results on the Lifshitz tails behavior of the integrated density of states for random magnetic Schr{ö}dinger operators, thereby providing the initial length-scale estimate, and a Wegner estimate, for such models.
We study the threshold behaviour of two dimensional Schr{\" o}dinger operators with finitely many local point interactions. We show that the resolvent can either be continuously extended up to the threshold, in which case we say that the operator is of regular type, or it has singularities associated with $s$ or p-wave resonances or even with an embedded eigenvalue at zero, for whose existence we give necessary and sufficient conditions. An embedded eigenvalue at zero may appear only if we have at least three centres. When the operator is of regular type we prove that the wave operators are bo…
Schr{ö}dinger eigenfunctions sharing the same modulus and applications to the control of quantum systems
本文研究薛定谔算子线性无关特征函数具有相同模的条件，并利用饱和性质与Duca-Nersesyan策略分析量子系统在能级间任意快速操控的可能性，同时将结果推广到量子图。
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…
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