On The Discrete Spectra of Schrödinger-Type Operators on one Dimensional Lattices
Annotatsiya
We consider a family $$\widehat{H}_{\mu}=\widehat{H}_{0}+\mu\widehat{V},\quad\mu>0,$$ of Schrödinger-type operators on the one dimensional lattice $$\mathbb{Z},$$ where $$\widehat{H}_{0}$$ is a Laurent–Toeplitz-type convolution operator with a given Hopping matrix $${\mathfrak{\hat{e}}}$$ and $$\widehat{V}$$ is a potential taking into account only the zero-range and one-range interactions, i.e., a 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 conditions on the regularity of $${\mathfrak{\widehat{e}}}$$ we completely describe the discrete spectrum of $$\widehat{H}_{\mu}$$ lying above the essential spectrum and study the dependence of eigenvalues on parameters $$\mu,$$ $$a$$ and $$b.$$