Unidirectionality and Husimi functions in constant-width neutrino billiards

We investigate the spectral properties and Husimi functions of relativistic quantum billiards (QBs) consisting of a spin-1/2 particle governed by the Dirac equation and confined to a planar domain of constant-width (CW) by imposing boundary conditions (BCs) on the spinor components. We consider those of neutrino billiards (NBs) proposed in (Berry and Mondragon 1987 Proc. R. Soc. A 412 53). The classical dynamics of billiards of corresponding shape is predominantly chaotic. CW billiards attracted particular attention because they exhibit unusual properties. Their classical dynamics features unidirectionality, whereas in the corresponding nonrelativistic QB a change of the rotational direction of motion is possible via dynamical tunneling, and the spectral properties coincide with those of typical quantum systems with violated time-reversal invariance. Unidirectionality of the quantum dynamics would arise in the structure of the Husimi functions. We analyze them for two realizations of CW NBs in the ultra-relativistic, i.e. the massless case and for massive cases and come to the result, that the modes can be separated into clockwise and counterclockwise modes and dynamical tunneling is absent. This is attributed to the BCs and the unidrectionality of the local current arising from them.