Resolvent estimates for one-dimensional Dirac operators with imaginary potentials

Abstract

We investigate massive one-dimensional Dirac operators perturbed by diagonal matrix potentials of the form $i V$ where the function $V$ is real-valued and unbounded at infinity. For such operators we find an $L^2$-realization with non-empty resolvent set using generalized coercivity and Schur complement dominance techniques. In the prototypical Airy-Dirac case $V(x)=x$, $x \in \mathbb{R}$, we derive the precise asymptotic behavior of the resolvent as the spectral parameter tends to infinity and also as the mass $m$ tends to $0$. Finally, we find the asymptotics of the resolvent norm for general potentials $V$ in terms of the Airy-Dirac resolvent, which in particular yields an asymptotic shape of $\varepsilon$-pseudospectral curves and establishes the optimality of the pseudospectral region found in [32].