Abstract
We study the Agmon estimate and the exponential decay of eigenfunctions for multi-dimensional discrete Schrödinger operators with emphasis on the microlocal analysis on the torus. We first consider the semiclassical setting where semiclassical continuous Schrödinger operators are discretized with the mesh width proportional to the semiclassical parameter. For a general class of potentials, we prove the Agmon estimate for eigenfunctions using the Agmon metric which is a Finsler metric rather than a Riemannian metric. Klein and Rosenberger (2008) proved this by a different argument in the case of a potential minimum. Our argument seems to be simpler. We also prove the Agmon estimate and the optimal anisotropic exponential decay of eigenfunctions for discrete Schrödinger operators in the non-semiclassical standard setting.
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