Maximal scarring for eigenfunctions of quantum graphs

TL;DR

This paper demonstrates the existence of highly localized eigenstates, called scars, on quantum star graphs with specific scattering matrices, showing they are maximal and occur under generic conditions.

Contribution

It proves the existence of maximal scarred eigenstates on quantum star graphs with specific scattering matrices, including Fourier and non-back-scattering types.

Findings

Existence of scarred eigenstates on star graphs with Fourier or non-back-scattering matrices.

Scarred states are maximally localized, with minimal entropy.

Scarred eigenstates exist for almost all length configurations.

Abstract

We prove the existence of scarred eigenstates for star graphs with scattering matrices at the central vertex which are either a Fourier transform matrix, or a matrix that prohibits back-scattering. We prove the existence of scars that are half-delocalised on a single bond. Moreover we show that the scarred states we construct are maximal in the sense that it is impossible to have quantum eigenfunctions with a significantly lower entropy than our examples. These scarred eigenstates are on graphs that exhibit generic spectral statistics of random matrix type in the large graph limit, and, in contrast to other constructions, correspond to non-degenerate eigenvalues; they exist for almost all choices of lengths.