We study the spectral properties of the Schrödinger-type operator${\hat{\mathit{H}}}_{\mu }:={\hat{\mathit{H}}}_0+\mu \hat{\mathit{V}},\mu \ge 0,$ associated to a one-particle system in d-dimensional lattice ${\mathbb{Z}}^d$, $d=1,2$, where the non-perturbed operator ${\hat{\mathit{H}}}_0$ is a self-adjoint convolution-type operator generated by a Hopping matrix $\hat{\mathfrak{e}}:{\mathbb{Z}}^d\to \mathbb{C}$ and the potential $\hat{\mathit{V}}$ is the multiplication operator by $\hat{v}:{\mathbb{Z}}^d\to \mathbb{R}$. Under certain regularity assumption on $\hat{\mathfrak{e}}$ and a decay assumption on $\hat{v}$, we establish the existence or non-existence and also the finiteness of eigenvalues of ${\hat{\mathit{H}}}_{\mu }$. Moreover, in the case of existence we study the asymptotics of eigenvalues of ${\hat{\mathit{H}}}_{\mu }$ as $\mu \searrow 0$.

中文翻译：

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一维和二维格上薛定ding型算子的束缚态

我们研究Schrödinger型算子的光谱性质${\hat{\mathit{H}}}_{\mu }：={\hat{\mathit{H}}}_0+\mu \widehat{\mathit{伏特}}，\mu \ge 0，$关联于d维晶格中的单粒子系统${\mathbb{ž}}^d$， $d=\mathrm{1个}，\mathrm{2个}$，其中不受干扰的运算符 ${\hat{\mathit{H}}}_0$ 是由Hopping矩阵生成的自伴随卷积型算子 $\widehat{\mathfrak{Ë}}：{\mathbb{ž}}^d\to \mathbb{C}$ 和潜力 $\widehat{\mathit{伏特}}$ 是乘法运算符 $\hat{v}：{\mathbb{ž}}^d\to \mathbb{[R}$。在一定规律性假设下$\widehat{\mathfrak{Ë}}$ 以及关于的衰减假设 $\hat{v}$，我们确定存在或不存在以及特征值的有限性 ${\hat{\mathit{H}}}_{\mu }$。此外，在存在的情况下，我们研究的特征值的渐近性${\hat{\mathit{H}}}_{\mu }$ 作为 $\mu \searrow 0$。