关于二维格上Schrödinger型算子的谱
On the spectrum of Schrödinger-type operators on two dimensional lattices
摘要：
我们考虑二维格点$\mathbb{Z}^2,$上的薛定谔型算子族$$ \widehat H_{a,b}(μ)=\widehat H_0 +μ\widehat V_{a,b}\quad μ>0, $$，其中$\widehat H_0$是具有给定跳跃矩阵$\hat{e}$的Laurent-Toeplitz型卷积算子，而$\widehat V_{a,b}$是只考虑零程和一程相互作用的势，即，一个乘法算子由一个功能$\hat v$，使$\hat v(0)=a,$ $\hat v(x)=b$为$|x|=1$和$\hat v(x)=0$为$|x|\ge2,$其中$a,b\in\mathbb{R}\setminus\{0\}.$在一定的条件下，对$\hat{e}$的正则性，我们完全描述的离散谱$\hat H_{a,b}(μ)$位于本质谱和研究的依赖关系的特征值参数$μ,$ $a$和$b.$此外，我们的阈值特征函数和共振。
英文摘要：
We consider a family $$ \widehat H_{a,b}(μ)=\widehat H_0 +μ\widehat V_{a,b}\quad μ>0, $$ of Schrödinger-type operators on the two dimensional lattice $\mathbb{Z}^2,$ where $\widehat H_0$ is a Laurent-Toeplitz-type convolution operator with a given Hopping matrix $\hat{e}$ and $\widehat V_{a,b}$ is a potential taking into account only the zero-range and one-range interactions, i.e., a multiplication operator by a function $\hat v$ such that $\hat v(0)=a,$ $\hat v(x)=b$ for $|x|=1$ and $\hat v(x)=0$ for $|x|\ge2,$ where $a,b\in\mathbb{R}\setminus\{0\}.$ Under certain conditions on the regularity of $\hat{e}$ we completely describe the discrete spectrum of $\hat H_{a,b}(μ)$ lying above the essential spectrum and study the dependence of eigenvalues on parameters $μ,$ $a$ and $b.$ Moreover, we characterize the threshold eigenfunctions and resonances.
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