An Agmon estimate for Schrödinger operators on Graphs

Abstract

The Agmon estimate shows that eigenfunctions of Schrödinger operators, $-\Delta \varphi +V\varphi =E\varphi$, decay exponentially in the `classically forbidden' region where the potential exceeds the energy level $\left\{x:V(x)>E\right\}$. Moreover, the size of $|\varphi (x)|$ 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.