论文标题
在Schrödinger-type操作员的光谱上
On the spectrum of Schrödinger-type operators on two dimensional lattices
论文作者
论文摘要
我们考虑一个家庭$$ \ widehat h_ {a，b}（μ）= \ widehat h_0 +μ\ widehat v_ {a，b} \quadμ> 0，schrödinger-type操作员在二维的lattice $ \ mathbb {z} $ {z} $ \ y $ \ s $ \ s $ \ s $ \ s $ \ y $ Laurent-toeplitz-type卷积操作员，带有给定的跳跃矩阵$ \ hat {e} $和$ \ wideHat v_ {a，b} $仅考虑到零范围和单范围的交互，即乘以零范围的交互，即，乘以函数$ \ hat $ \ hat $ \ hat $ \ hat $ \ hat $ \ hat $ \ hat $ \ hat $ \ f（$ \ hat） $ | x | = 1 $和$ \ hat v（x）= 0 $ for $ | x | \ ge2，$ a，$ a，b \ in \ mathbb {r} \ setMinus \ {0 \}。并研究特征值对参数的依赖性$μ，$ a $ and $ b。$此外，我们表征了阈值特征函数和共振。
We consider a family $$ \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.
