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The spectra of massless Dirac operators are of essential interest e.g. forthe electronic properties of graphene, but fundamental questions such as theexistence of spectral gaps remain open. We show that the eigenvalues ofmassless Dirac operators with suitable real-valued potentials lie inside smallsets easily characterised in terms of properties of the potentials, and weprove a Schnol'-type theorem relating spectral points to polynomial boundednessof solutions of the Dirac equation. Moreover, we show that, under minimalhypotheses which leave the potential essentially unrestrained in large parts ofspace, the spectrum of the massless Dirac operator covers the whole real line;in particular, this will be the case if the potential is nearly constant in asequence of regions.

Schnol's theorem and spectral properties of massless Dirac operators with scalar potentials