An Agmon estimate for Schrödinger operators on Graphs

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