On the Number and Locations of Eigenvalues of the Discrete Schrödinger Operator on a Lattice
Аннотация
We consider the family of Schrödinger operators $$\hat{H}_{\gamma\lambda\mu}=\hat{H}_{0}+\hat{V}_{\gamma\lambda\mu}$$ on the one-dimensional lattice $$\mathbb{Z}$$ , where $$\hat{H}_{0}$$ is a convolution operator with a given Hopping matrix $$\hat{\varepsilon}$$ , and $$\hat{V}_{\gamma\lambda\mu}$$ is a multiplication operator by the function $$\hat{v}$$ such that $$\hat{v}(0)=\gamma$$ , $$\hat{v}(x)=\frac{\lambda}{2}$$ for $$|x|=1$$ , $$\hat{v}(x)=\frac{\mu}{2}$$ for $$|x|=2$$ and $$\hat{v}(x)=0$$ for $$|x|>2$$ , $$\gamma,\lambda,\mu\in\mathbb{R}$$ . Under certain conditions on the potential, we prove that the discrete Schrödinger operator $$\hat{H}_{\gamma\lambda\mu}$$ can have zero, one, two or three eigenvalues outside the essential spectrum. Moreover, we obtain conditions for the existence of three eigenvalues, two of them situated below the bottom of the essential spectrum and the other one above its top.