An Agmon estimate for Schr\"odinger operators on Graphs

TL;DR

This paper extends the Agmon estimate, which describes exponential decay of eigenfunctions in forbidden regions, to Schr"odinger operators on graphs by defining an explicit Agmon metric and proving decay estimates.

Contribution

It introduces an Agmon estimate for Schr"odinger operators on graphs, including an explicit Agmon metric and decay bounds for eigenfunctions.

Findings

Established an Agmon estimate for graph Schr"odinger operators.

Defined an explicit Agmon metric on graphs.

Proved pointwise decay estimates for eigenfunctions.

Abstract

The Agmon estimate shows that eigenfunctions of Schr\"odinger 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.