An Agmon estimate for Schrödinger operators on graphs

The Agmon estimate shows that eigenfunctions of Schrödinger operators, -Δ ϕ +V ϕ =E ϕ , decay exponentially in the `classically forbidden' region where the potential exceeds the energy level {x :V (x )>E }. 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.