The Existence and Asymptotics of Eigenvalues of Schrödinger Operator on Two Dimensional Lattices

We study the spectral properties of the Schrödinger-type operator $$\widehat{H}_{\mu}:=\widehat{H}_{0}+\mu\widehat{V}$$ , $$\mu>0,$$ associated to a one-particle system in two dimensional lattice $${\mathbb{Z}}^{2},$$ where the non-perturbed operator $$\widehat{H}_{0}$$ is a convolution-type operator generated by a Hopping matrix $$\widehat{\varepsilon}:{\mathbb{Z}}^{d}\to{\mathbb{C}}$$ and the potential $$\widehat{V}$$ is the multiplication operator by a function $$\widehat{v}$$ such that $$\widehat{v}(0)=a$$ , $$\widehat{v}(x)=b$$ for $$|x|=1$$ and $$\widehat{v}(x)=0$$ for $$|x|\geq 2,$$ where $$a,b\in{\mathbb{R}}\setminus\{0\}.$$ Under certain regularity assumption on $$\widehat{\varepsilon},$$ we establish the existence or non-existence of eigenvalues of $$\widehat{H}_{\mu}$$ lying below the essential spectrum. Moreover, in the case of existence we study the convergent expansion of eigenvalues for sufficiently small and positive $$\mu.$$

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