Abstract. We show that the non-embedded eigenvalues of the Dirac operator on the real line with non-Hermitian potential V lie in the disjoint union of two disks in the right and left half plane, respectively, provided that the L 1 -norm of V is bounded from above by the speed of light times the reduced Planck constant. An analogous result for the Schrödinger operator, originally proved by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For massless Dirac operators, the condition on V implies the absence of nonreal eigenvalues. Our results are further generalized to potentials with slower decay at infinity. As an application, we determine bounds on resonances and embedded eigenvalues of Dirac operators with Hermitian dilation-analytic potentials.

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Abstract. We investigate the effect of non-symmetric relatively bounded perturbations on the spectrum of self-adjoint operators. In particular, we establish stability theorems for one or infinitely many spectral gaps along with corresponding resolvent estimates. These results extend, and improve, classical perturbation results by Kato and by Gohberg/Kreȋn. Further, we study essential spectral gaps and perturbations exhibiting additional structure with respect to the unperturbed operator; in the latter case, we can even allow for perturbations with relative bound ≥ 1. The generality of our results is illustrated by several applications, massive and massless Dirac operators, point-coupled periodic systems, and two-channel Hamiltonians with dissipation.

We provide examples of operators T (D) + V with decaying potentials that have embedded eigenvalues. The decay of the potential depends on the curvature of the Fermi surfaces of constant kinetic energy T . We make the connection to counterexamples in Fourier restriction theory.

Abstract. We derive bounds on the location of non-embedded eigenvalues of Dirac operators on the half-line with non-Hermitian L 1 -potentials. The results are sharp in the non-relativistic or weak-coupling limit. In the massless case, the absence of discrete spectrum is proved under a smallness assumption.