Abstract

The Agmon estimate shows that eigenfunctions of Schrödinger operators, $$ -\Delta \phi + V \phi = E \phi $$ , decay exponentially in the ‘classically forbidden’ region where the potential exceeds the energy level $$\left\{ x: V(x) > E \right\} $$ . Moreover, the size of $$|\phi (x)|$$ is bounded in terms of a weighted (Agmon) distance between x and the allowed region. We derive such a statement on graphs when $$-\Delta $$ is replaced by the graph Laplacian $$L = D-A$$ : we identify an explicit Agmon metric and prove a pointwise decay estimate in terms of the Agmon distance.

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