An Agmon estimate for Schrödinger operators on Graphs

The Agmon estimate shows that eigenfunctions of Schrödinger operators, $-Δϕ+ V ϕ= E ϕ$, decay exponentially in the `classically forbidden' region where the potential exceeds the energy level $\left\{x: V(x) > E \right\}$. Moreover, the size of $|ϕ(x)|$ is bounded in terms of a weighted (Agmon) distance between $x$ and the allowed region. We derive such a statement on graphs when $-Δ$ 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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