The Spectrum of a Non-local Discrete Schrödinger Operator with a Delta Potential on the One-Dimentional Lattice
Abstract
In this work, we consider a non-local discrete Schrödinger operator of the third order $$\hat{h}_{\mu}=\hat{\Delta}\hat{\Delta}\hat{\Delta}-\mu\hat{\delta}{[x,0]}$$ in a one-dimensional lattice $$\mathbb{Z}$$ . Here $$\hat{\Delta}$$ is the discrete Laplacian operator, and $$\hat{\delta}{[x,0]}$$ is the delta potential of a concentrated Kroneker at zero. It has been proved that there is a single eigenvalue to the left of the essential spectrum for any $$\mu>0$$ and to the right of the essential spectrum for $$\mu<0$$ . In addition, the analytical function $$z(\mu)$$ was constructed as a function of $$\mu$$ around the point $$\gamma>0$$ $$(0,\gamma)$$ right and $$(-\gamma,0)$$ left of the point $$\mu=0$$ , in particular, the asymptotes of $$z(\mu)$$ as $$\mu\nearrow 0$$ and $$\mu\searrow 0$$ are found.