Quantum Return Probability for Substitution Potentials

TL;DR

This paper introduces an effective exponent to describe the decay of quantum return probability in discrete Schrödinger operators, analyzing various substitution potentials with differing randomness levels.

Contribution

It proposes a new exponent governing decay behavior and investigates its relation to the spectral type of substitution sequences, revealing unexpected correlations.

Findings

More random sequences have smaller decay exponents.

The decay exponent does not fully align with the spectral type of the sequences.

The study provides insights into quantum dynamics in non-periodic potentials.

Abstract

We propose an effective exponent ruling the algebraic decay of the average quantum return probability for discrete Schrodinger operators. We compute it for some non-periodic substitution potentials with different degrees of randomness, and do not find a complete qualitative agreement with the spectral type of the substitution sequences themselves, i.e., more random the sequence smaller such exponent.