Threshold Analysis of the One-Rank Perturbation Non-Local Discrete Laplacian

The behaviour of the embedded eigenvalues and resonances is discussed at the lower threshold of the essential spectrum of non-local discrete Schrödinger operators with the Kroneker $$\delta$$ -potential with the mass $$\mu\geq 0$$ . This operator is constructed by taking a strictly increasing $$C$$ function of the standard discrete Laplacian instead of the original one. The dependence of the existence of resonances on this function and the lattice dimension are explicitly derived. We study the limits of eigenvalues as $$\mu\nearrow+\infty$$ and $$\mu\searrow\mu_{0}$$ , where $$\mu_{0}$$ is the value of $$\mu$$ , which provides there existence of the threshold resonance.

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