Counting Schrödinger boundstates: semiclassics and beyond

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。

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…

三维次黎曼本征函数的渐近正则性：周期情形

Asymptotic regularity of sub-Riemannian eigenfunctions in dimension 3: the periodic case

本文研究常非零曲率紧黎曼曲面单位切丛上与次黎曼拉普拉斯算子相关的半经典Schrödinger算子，证明周期Reeb流下高频本征函数的额外正则性，并应用于推导磁Schrödinger算子低能本征模的正则性。

A semiclassical Birkhoff normal form for symplectic magnetic wells

In this paper we construct a Birkhoff normal form for a semiclassical magnetic Schr{ö}dinger operator with non-degenerate magnetic field, and discrete magnetic well, defined on an even dimensional riemannian manifold M. We use this normal form to get an expansion of the first eigenvalues in powers of h^{1/2}, and semiclassical Weyl asymptotics for this operator.

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。

费米子半经典Lp估计

Fermionic semiclassical Lp estimates

该研究受囚禁势中自由费米子浓度问题驱动，将带约束势薛定谔算子的半经典Lp估计推广到密度矩阵，借助半经典与多体工具证明，给出谱簇界并讨论了界的最优性。