Quantum magnetic billiards: boundary conditions and gauge transformations

Abstract

We study admissible boundary conditions for a charged quantum particle in a two-dimensional region subjected to an external magnetic field, i.e. a quantum magnetic billiard. After reviewing some physically interesting classes of admissible boundary conditions (magnetic Robin and chiral boundary conditions), we turn our attention to the role of gauge transformations in a magnetic billiard: in particular, we introduce gauge covariant boundary conditions, and find a sufficient condition for gauge covariance which is satisfied by all the aforementioned examples.

Hearing the shape of a quantum boundary condition

“…This work could further be expanded into different directions: by considering an analogous problem in higher-dimensional systems (which we expect to be a non-trivial task 36,37 ), as e.g. in a billiard, 38 or by moving to more complex onedimensional systems such as quantum graphs, or also by considering a relativistic setting described by the Dirac operator. In any case we believe that our results shed some light on the isospectrality problem of a "quantum drum" from a new point of view, which focuses on the boundary conditions rather than on the shape of the drum, and more generally contribute to the very active topic of boundary effects and their interplay with the bulk of a quantum system.…”