A numerical method is presented which allows to compute the spectrum of the Schroedinger operator for a particle constrained on a two dimensional flat torus under the combined action of a transverse magnetic field and any conservative force. The meth
od employs a fast Fourier transform to accurately represent the momentum variables and takes into account the twisted boundary conditions required by the presence of the magnetic field. An accuracy of twelve digits is attained even with coarse grids. Landau levels are reproduced in the case of a uniform field satisfying Diracs condition. A new fine structure of levels within the single Landau level is formed when the field has a sinusoidal component with period commensurable to the integer magnetic charge.
The S-matrix in gravitational high energy scattering is computed from the region of large impact parameters b down to the regime where classical gravitational collapse is expected to occur. By solving the equation of an effective action introduced by
Amati, Ciafaloni and Veneziano we find that the perturbative expansion around the leading eikonal result diverges at a critical value signalling the onset of a new regime. We then discuss the main features of our explicitly unitary S-matrix down to the Schwarzschilds radius R=2G s^(1/2), where it diverges at a critical value b ~ 2.22 R of the impact parameter. The nature of the singularity is studied with particular attention to the scaling behaviour of various observables at the transition. The numerical approach is validated by reproducing the known exact solution in the axially symmetric case to high accuracy.
