论文标题
离散的schrödinger运营商的界限在一个和二维晶格上
Bound states of discrete Schrödinger operators on one and two dimensional lattices
论文作者
论文摘要
我们研究离散schrödinger操作员$$ \ widehath_μ= \ widehat h_0 +μ\ widehat {v}，\qquadμ\ ge0，$ ge0，$ d $ d $ d $ d $ dattice lattice $ \ mathbb {z z}^d = d = d = d = d = d = d = 1 d = d = d = d = d = d = d = d = d = d = d = d = 1 $ \ hat h_0 $是由$ \ hat e生成的自动化laurent toeplitz-type操作员：在$ \ hat e $上的假设以及对$ \ hat v $的衰减假设，我们确定了$ \ hath_μ的特征值的存在或不存在的存在，也是$ \h_μ。
We study the spectral properties of discrete Schrödinger operator $$ \widehat H_μ=\widehat H_0 + μ\widehat{V},\qquad μ\ge0, $$ associated to a one-particle system in $d$-dimensional lattice $\mathbb{Z}^d, $ $d=1,2,$ where the non-perturbed operator $\hat H_0$ is a self-adjoint Laurent-Toeplitz-type operator generated by $\hat e:\mathbb{Z}^d\to\mathbb{C}$ and the potential $\hat V$ is the multiplication operator by $\hat v:\mathbb{Z}^d\to\mathbb{R}.$ Under certain regularity assumption on $\hat e$ and a decay assumption on $\hat v$, we establish the existence or non-existence and also the finiteness of eigenvalues of $\hat H_μ.$ Moreover, in the case of existence we study the asymptotics of eigenvalues of $\hat H_μ$ as $μ\searrow 0.$
