Dispersive decay estimates for Dirac equations with a domain wall

TL;DR

This paper establishes dispersive decay estimates for a family of one-dimensional Dirac Hamiltonians with domain walls, revealing how threshold resonances affect decay rates, and is the first to do so for non-compact perturbations.

Contribution

It provides the first dispersive decay estimates for Dirac operators that are not relatively compact perturbations of free operators, explicitly showing the impact of threshold resonances.

Findings

Decay rates vary with the phase-shift parameter $ au$

Threshold resonances influence the decay behavior

Results apply to topologically protected quantum states

Abstract

Dispersive time-decay estimates are proved for a one-parameter family of one-dimensional Dirac Hamiltonians with dislocations; these are operators which interpolate between two phase-shifted massive Dirac Hamiltonians at $x=+\infty$ and $x=-\infty$. This family of Hamiltonians arises in the theory of topologically protected states of one-dimensional quantum materials. For certain values of the phase-shift parameter, $\tau$, the Dirac Hamiltonian has a {\it threshold resonance} at the endpoint of its essential spectrum. Such resonances are known to influence the time-decay rate. Our main result explicitly displays the transition in time-decay rate as $\tau$ varies between resonant and non-resonant values. Our results appear to be the first dispersive time-decay estimates for Dirac Hamiltonians which are not a relatively compact perturbation of a free Dirac operator.