Threshold Laws for the Break-up of Atomic Particles into Several Charged Fragments

The processes with three or more charged particles in the final state exhibit particular threshold behavior, as inferred by the famous Wannier law for (2e + ion) system. We formulate a general solution which determines the threshold behavior of the cross section for multiple fragmentation. Applications to several systems of particular importance with three, four and five leptons (electrons and positrons) in the field of charged core; and two pairs of identical particles with opposite charges are presented. New threshold exponents for these systems are predicted, while some previously suggested threshold laws are revised.Comment: 40 pages, Revtex, scheduled for the July issue of Phys.Rev.A (1998

Quantum Disorder and Quantum Chaos in Andreev Billiards

We investigate the crossover from the semiclassical to the quantum description of electron energy states in a chaotic metal grain connected to a superconductor. We consider the influence of scattering off point impurities (quantum disorder) and of quantum diffraction (quantum chaos) on the electron density of states. We show that both the quantum disorder and the quantum chaos open a gap near the Fermi energy. The size of the gap is determined by the mean free time in disordered systems and by the Ehrenfest time in clean chaotic systems. Particularly, if both times become infinitely large, the density of states is gapless, and if either of these times becomes shorter than the electron escape time, the density of states is described by random matrix theory. Using the Usadel equation, we also study the density of states in a grain connected to a superconductor by a diffusive contact.Comment: 20 pages, 10 figure

Logarithmic contribution to the density of states of rectangular Andreev billiards.

We demonstrate that the exact quantum mechanical calculations are in good agreement with the semiclassical predictions for rectangular Andreev billiards, and therefore for a large number of open channels it is sufficient to investigate the Bohr-Sommerfeld approximation of the density of states . We present exact calculations of the classical path length distribution P(s), which is a nondifferentiable function of s, but whose integral is a smooth function with logarithmically dependent asymptotic behavior. Consequently, the density of states of rectangular Andreev billiards has two contributions on the scale of the Thouless energy: one which is well-known and is proportional to the energy, and the other which shows a logarithmic energy dependence. It is shown that the prefactors of both contributions depend on the geometry of the billiards but have universal limiting values when the width of the superconductor tends to zero