Counterexample to the Bohigas Conjecture for Transmission Through aOne-Dimensional Lattice

Abstract

Resonances in particle transmission through a 1D finite lattice are studied in the presence of a finite number of impurities. Although this is a one-dimensional system that is classically integrable and has no chaos, studying the statistical properties of the spectrum such as the level spacing distribution and the spectral rigidity shows quantum chaos signatures. Using a dimensionless parameter that reflects the degree of state localization, we demonstrate how the transition from regularity to chaos is affected by state localization. The resonance positions are calculated using both the Wigner-Smithtime-delay and a Siegert state method, which are in good agreement. Our results give evidence for the existence of quantum chaos in one dimension which is a counter-example to the Bohigas-Giannoni-Schmit conjecture.