Spectral localization for semimetals and Callias operators

A semiclassical argument is used to show that the low-lying spectrum of a self-adjoint operator, the so-called spectral localizer, determines the number of Dirac or Weyl points of an ideal semimetal. Apart from the ion-mobility spectrometer localization procedure, an explicit computation for the local toy models given by a Dirac or Weyl point is the key element of proof. The argument has numerous similarities to Witten’s reasoning leading to the strong Morse inequalities. The same techniques allow to prove a spectral localization for Callias operators associated with potentials with isolated gap-closing points.