Quantum lattice transport along an infinitely extended perturbation

TL;DR

This paper investigates how an infinite perturbation affects the spectral properties of a periodic quantum lattice, revealing conditions for spectrum preservation and localization probabilities.

Contribution

It introduces a detailed analysis of spectral changes in a quantum lattice due to localized vertex perturbations, including conditions for spectrum preservation and localization probabilities.

Findings

Spectrum remains unchanged if perturbation strength increases.

Additional spectral bands appear under certain perturbation conditions.

Probability of localized states near perturbation is 50%.

Abstract

We consider a periodic quantum graph in the form of a rectangular lattice with the $\delta$-coupling of strength $\gamma$ in the vertices perturbed by changing the latter at an infinite straight array of vertices to a $\widetilde\gamma\ne\gamma$. We analyze the band spectrum of the system and show that it remains preserved as a set provided $\widetilde\gamma>\gamma>0$ while for all the other combinations additional band appear in some or all gaps of the unperturbed system. We also prove that for a randomly chosen positive energy, the probability of existence of a state exponentially localized in the vicinity of the perturbation equals $\frac12$.