Abstract

The one-dimensional massless Dirac operator not only does not have a central gap in its spectrum, but it is even unitarily equivalent to the free massless operator and hence has purely absolutely continuous spectrum covering the whole real line.

The talk reports on work regarding the question to what extent similar general statements about the essential spectrum and the absolutely continuous spectrum are possible in the higher-dimensional case. Under the assumption of spherical symmetry, it can be shown that the spectrum is purely absolutely continuous outside the limit range of the potential, thus giving an a priori set containing all eigenvalues. This can be partly extended to a far more general situation by a virial technique. Moreover, the essential spectrum is the whole real line for a large class of potentials satisfying a local separability condition. This is joint work with T. Umeda.