Quasiclassical mass spectrum of the black hole model with selfgravitating dust shell

We consider a quantum mechanical black hole model introduced in {\it Phys.Rev.}, {\bf D57}, 1118 (1998) that consists of the selfgravitating dust shell. The Schroedinger equation for this model is a finite difference equation with the shift of the argument along the imaginary axis. Solving this equation in quasiclassical limit in complex domain leads to quantization conditions that define discrete quasiclassical mass spectrum. One of the quantization conditions is Bohr-Sommerfeld condition for the bound motion of the shell. The other comes from the requirement that the wave function is unambiguously defined on the Riemannian surface on which the coefficients of Schroedinger equation are regular. The second quantization condition remains valid for the unbound motion of the shell as well, and in the case of a collapsing null-dust shell leads to $m\sim\sqrt{k}$ spectrum.Comment: 35 pages, 8 figures, to appear in Phys. Rev.

Nonlinear σ\sigma model for disordered superconductors

We suggest a novel nonlinear $\sigma$-model for the description of disordered superconductors. The main distinction from existing models lies in the fact that the saddle point equation is solved non-perturbatively in the superconducting pairing field. It allows one to use the model both in the vicinity of the metal-superconductor transition and well below its critical temperature with full account for the self-consistency conditions. We show that the model reproduces a set of known results in different limiting cases, and apply it for a self-consistent description of the proximity effect at the superconductor-metal interface.Comment: Revised version, 8 pages, 1 fig., revtex; final version, as published, contains a few corrections in the summar