Absolutely continuous spectrum for a random potential on a tree with strong transverse correlations and large weighted loops

Absolutely continuous spectrum for Laplacians on radial metric trees and periodicity

On an infinite, radial metric tree graph we consider the corresponding Laplacian equipped with self-adjoint vertex conditions from a large class including $δ$- and weighted $δ'$-couplings. Assuming the numbers of different edge lengths, branching numbers and different coupling conditions to be finite, we prove that the presence of absolutely continuous spectrum implies that the sequence of geometric data of the tree as well as the coupling conditions are eventually periodic. On the other hand, we provide examples of self-adjoint, non-periodic couplings which admit absolutely continuous spectru…

Scattering the geometry of weighted graphs

Given two weighted graphs $(X,b_k,m_k)$, $k=1,2$ with $b_1\sim b_2$ and $m_1\sim m_2$, we prove a weighted $L^1$-criterion for the existence and completeness of the wave operators $ W_{\pm}(H_{2},H_1, I_{1,2})$, where $H_k$ denotes the natural Laplacian in $\ell^2(X,m_k)$ w.r.t. $(X,b_k,m_k)$ and $I_{1,2}$ the trivial identification of $\ell^2(X,m_1)$ with $\ell^2(X,m_2)$. In particular, this entails a very general criterion for the absolutely continuous spectra of $H_1$ and $H_2$ to be equal.

The limiting absorption principle for the discrete Wigner-von Neumann operator

We apply weighted Mourre commutator theory to prove the limiting absorption principle for the discrete Schr{ö}dinger operator perturbed by the sum of a Wigner-von Neumann and long-range type potential. In particular, this implies a new result concerning the absolutely continuous spectrum for these operators even for the one-dimensional operator. We show that methods of classical Mourre theory based on differential inequalities and on the generator of dilation cannot apply to the mentionned Schr{ö}dinger operators.

具有衰减势的离散多维薛定谔算子的弹道输运

Ballistic Transport for Discrete Multi-Dimensional Schrödinger Operators With Decaying Potential

研究具有衰减势的离散薛定谔算子的弹道输运性质，利用交换子方法和Mourre估计证明了绝对连续谱上的加权ℓ²范数以t^r速率增长。

为什么 Meyer 集会产生衍射？

Why do Meyer sets diffract?

本文研究了 Meyer 集衍射现象的普适性，构造了弱模型集所有子集共享的 Bragg 峰与范数几乎周期集合，并将结果推广到 Meyer 集上的一般加权 Dirac 梳及广义 Eberlein 分解。