On the spectrum of Schr\"odinger-type operators on two dimensional lattices

We consider a family $$\widehat H_{a,b}(\mu)=\widehat H_0 +\mu \widehat V_{a,b}\quad \mu>0,$$ of Schr\"odinger-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 $\hat H_{a,b}(\mu)$ lying above the essential spectrum and study the dependence of eigenvalues on parameters $\mu,$ $a$ and $b.$ Moreover, we characterize the threshold eigenfunctions and resonances.