Absolute continuity and spectral concentration for slowly decaying potentials

Zero energy asymptotics of the resolvent for a class of slowly decaying potentials

We prove a limiting absorption principle at zero energy for two-body Schrödinger operators with long-range potentials having a positive virial at infinity. More precisely, we establish a complete asymptotic expansion of the resolvent in weighted spaces when the spectral parameter varies in cones; one of the two branches of boundary for the cones being given by the positive real axis. The principal tools are absence of eigenvalue at zero, singular Mourre theory and microlocal estimates.

一维Schrödinger算子在指数衰减扰动下的谱测度的Hölder连续性与傅里叶渐进行为

Hölder continuity and Fourier asymptotics of spectral measures for 1D Schrödinger operators under decaying perturbations

本文研究了一维Schrödinger算子在指数衰减扰动下的谱测度连续性与量子返回概率渐进行为，证明了谱测度的1/2-Hölder连续性和纯绝对连续谱的持续性。

具有缓慢衰减势的分数阶 Schrödinger 算子的局部时间衰减及经典禁戒区域中的弱 Agmon 型估计

Local time decay for fractional Schrödinger operators with slowly decaying potentials and a weaker Agmon type estimate in a classically forbidden region

本文研究了具缓慢衰减正势的分数阶 Schrödinger 算子的局部时间衰减，利用奇异符号演算在经典禁戒区域建立弱 Agmon 估计，证明了预解式在零附近光滑及时间传播子具有快速局部时间衰减，并得出 Riesz 算子是具奇异符号的伪微分算子。

Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: some optimal results

The absolutely continuous spectrum of one-dimensional Schrödinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectrum of free and periodic Schrödinger operators is preserved under all perturbations $V(x)$ satisfying $|V(x)|\leq C(1+x)^{-α}$, $α>\frac{1}{2}.$ This result is optimal in the power scale. More general classes of perturbing potentials which are not necessarily power decaying are also treated. A general criterion for stability of the absolutely continuous spectrum of one-dimensional Schr…

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…

Preservation of the absolutely continuous spectrum of Schrödinger equation under perturbations by slowly decreasing potentials and a.e. convergence of integral operators

We prove new criteria of stability of the absolutely continuous spectrum of one-dimensional Schrödinger operators under slowly decaying perturbations. As applications, we show that the absolutely continuous spectrum of the free and periodic Schrödinger operators is preserved under perturbations by all potentials $V(x)$ satisfying $|V(x)| \leq C(1+x)^{-\frac{2}{3}-ε}.$ The main new technique includes an a.e.\ convergence theorem for a class of integral operators.

Counting Schrödinger boundstates: semiclassics and beyond

This is a survey of the basic results on the behavior of the number of the eigenvalues of a Schrödinger operator, lying below its essential spectrum. We discuss both fast decaying potentials, for which this behavior is semiclassical, and slowly decaying potentials, for which the semiclassical rules are violated. Some new results are presented, concerning operators on product manifolds and graphs.

带有衰减扰动的周期CMV矩阵的定量谱稳定性与精确傅里叶估计

Quantitative spectral stability and sharp Fourier estimates for periodic CMV matrices with decaying perturbations

该研究针对Verblunsky系数为周期序列衰减扰动的半直线CMV矩阵，建立了扰动谱测度的Hölder连续性与精确傅里叶估计，为相关单位圆正交多项式（OPUC）研究提供了新结论。

Absolute continuity of the spectrum of a Schrodinger operator with a potential which is periodic in some directions and decays in others

We prove that the spectrum of a Schrodinger operator that is periodic in certain directions and super-exponentially decaying in the others is purely absolutely continuous.

Discrete spectrum of Schrödinger operators with oscillating decaying potentials

We consider the Schrödinger operator $H_{ηW} = -Δ+ ηW$, self-adjoint in $L^2({\mathbb R}^d)$, $d \geq 1$. Here $η$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. We study the asymptotic behaviour of the discrete spectrum of $H_{ηW}$ near the origin, and due to the irregular decay of $ηW$, we encounter some non semiclassical phenomena. In particular, $H_{ηW}$ has less eigenvalues than suggested by the semiclassical intuition.

Absolutely continuous spectrum of a one-paramentric family of Schrödinger operators

We consider a family of operators $-Δ+ t V$ with a slowly decaying and oscillating potential $V$. We prove that the absolutely continuous spectrum of this operator is essentially supported by $[0,\infty)$ for almost every $t$.

Absence of continuous spectral types for certain nonstationary random models

We consider continuum random Schrödinger operators of the type $H_ω = -Δ+ V_0 + V_ω$ with a deterministic background potential $V_0$. We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of $-Δ+V_0$. The models we treat include random surface potentials as well as sparse or slowly decaying random potentials. In particular, we establish absence of absolutely continuous surface spectrum for random potentials supported near a one-dimensional surface (``random tube'') in arbitrary dimension.