R\'enyi and Tsallis entropies related to eigenfunctions of quantum graphs

TL;DR

This paper investigates how eigenfunctions of finite quantum graphs are distributed using Rényi and Tsallis entropies, deriving bounds and exploring their relation to variances, with implications for quantum ergodicity.

Contribution

It introduces a new approach using generalized entropies to analyze eigenfunction distribution on quantum graphs, including bounds and relations to variances.

Findings

Lower bounds on symmetrized generalized entropies derived.

Relations between entropies and variances of eigenfunctions examined.

Specific analysis of star graph eigenfunctions conducted.

Abstract

For certain families of finite quantum graphs, we study the question of how eigenfunctions are distributed over the graph. To characterize properties of the distribution, generalized entropies of the R\'{e}nyi and Tsallis types are considered. The presented approach is similar to entropic uncertainty relations of the Maassen-Uffink type. Using the Riesz theorem, we derive lower bounds on symmetrized generalized entropies of eigenfunctions. A quality of such estimates will depend on boundary conditions used at vertices of the given graph. R\'{e}nyi and Tsallis entropies of eigenfunctions of star graphs are separately examined. Relations between generalized entropies and variances of eigenfunctions are considered as well. When such relations remain valid on average, they may be used in studies of quantum ergodicity.