No quantum ergodicity for star graphs

TL;DR

This paper demonstrates that eigenfunctions on star graphs with incommensurate bond lengths do not become uniformly distributed as the number of bonds increases, revealing non-ergodic quantum behavior and eigenfunction localization.

Contribution

It provides the first proof of non-quantum ergodicity for star graphs and constructs explicit eigenfunction subsequences that localize on pairs of bonds.

Findings

Eigenfunctions are not quantum ergodic on star graphs with incommensurate bond lengths.

Explicit construction of eigenfunction subsequences localizing on pairs of bonds.

Eigenfunction localization resembles scars on unstable periodic orbits.

Abstract

We investigate statistical properties of the eigenfunctions of the Schrodinger operator on families of star graphs with incommensurate bond lengths. We show that these eigenfunctions are not quantum ergodic in the limit as the number of bonds tends to infinity by finding an observable for which the quantum matrix elements do not converge to the classical average. We further show that for a given fixed graph there are subsequences of eigenfunctions which localise on pairs of bonds. We describe how to construct such subsequences explicitly. These constructions are analogous to scars on short unstable periodic orbits.