Spectral Properties of Continuum Fibonacci Schrödinger Operators

Exact observability properties of subelliptic wave and Schr{ö}dinger equations

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.

Limiting absorption principle for discrete Schr{ö}dinger operators with a Wigner-von Neumann potential and a slowly decaying potential

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…

Boundedness of Schr{ö}dinger operator in energy space

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.

环面上磁薛定谔方程的长时间动力学

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。

Inverse scattering at fixed energy for radial magnetic Schr{ö}dinger operators with obstacle in dimension two

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…

Scaling limits of bosonic ground states, from many-body to nonlinear Schr{ö}dinger

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…

Local $L^p$ norms of Schr{ö}dinger eigenfunctions on $\mathbb{S}^2$

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.

On Lieb--Thirring inequalities for one-dimensional non-self-adjoint Jacobi and Schr{\" o}dinger operators

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 …

区间上半经典 Schrödinger 特征函数的一致观测

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。

Observability and Controllability for the Schr{ö}dinger Equation on Quotients of Groups of Heisenberg Type

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…

共享相同模的薛定谔特征函数及其在量子系统控制中的应用

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。