Dispersive decay estimates for Dirac equations with a domain wall

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.