Anderson Localization for Time Quasi Periodic Random Schödinger and Wave Operators

Multiple wells Airy-Schr{ö}dinger operator and integrated density of states of the periodic Airy-Schr{ö}dinger operator

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…

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.

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 …

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…

Schrödinger 算子的谱流与 Levinson 定理

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。

紧曲面上带白噪声势的Strichartz不等式

Strichartz inequalities with white noise potential on compact surfaces

该研究借助高阶仿受控演算描述的Anderson哈密顿量H，证明了二维流形上带乘性噪声的薛定谔方程与波动方程的Strichartz不等式，并为相关非线性方程建立了低正则性解理论。

Two dimensional Schr{\" o}dinger operators with point interactions: threshold expansions, zero modes and $L^p$-boundedness of wave operators

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…

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。

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…