On Agmon metrics and exponential localization for quantum graphs

TL;DR

This paper studies how eigenfunctions decay on quantum graphs using Agmon's method, establishing bounds that depend on graph structure, with implications for understanding exponential localization.

Contribution

It introduces Agmon-based decay estimates for eigenfunctions on quantum graphs and compares generic and structure-specific decay rates, revealing conditions for faster decay.

Findings

Exponential decay rate is controlled by classical Liouville-Green estimates.

Additional graph structure can lead to more rapid eigenfunction decay.

Two alternative estimates are provided under restrictive assumptions.

Abstract

We investigate the rate of decrease at infinity of eigenfunctions of quantum graphs by using Agmon's method to prove $L^2$ and $L^\infty$ bounds on the product of an eigenfunction with the exponential of a certain metric. A generic result applicable to all graphs is that the exponential rate of decay is controlled by an adaptation of the standard estimates for a line, which are of classical Liouville-Green (WKB) form. Examples reveal that this estimate can be the best possible, but that a more rapid rate of decay is typical when the graph has additional structure. In order to understand this fact, we present two alternative estimates under more restrictive assumptions on the graph structure that pertain to a more rapid decay. One of these depends on how the eigenfunction is distributed along a particular chosen path, while the other applies to an average of the eigenfunction over edges at a given distance from the root point.