A Rellich type theorem for discrete Schrödinger operators (Q479861)

The Rellich theorem asserts that, if \(u \in H^2_{\mathrm{loc}}(\mathbb{R}^d)\) satisfies \((-\Delta-\lambda)u=0\) on \(|x|>R_0\) for some constants \(R_0, \lambda\) and \(u(x)=o(|x|^{-(d-1)/2})\) for \(|x| \to \infty\), then \(u(x)=0\) on \(|x|>R_0\), meaning the non-existence of eigenvalues for the operator. The present paper extends this theorem to the discrete Laplacian on square lattices and shows the non-existence of eigenvalues for discrete Schrödinger operators.