Arbitrary Potential

M. G. Rozman, P. Reineker, R. Tehver

A recurrence method for analytical and numerical evaluation of tunneling, transmission, and reflection amplitudes is developed. As the first step, a rule for composition of two arbitrary scatterers separated by a region of constant potential is obtained. Transmission and reflection amplitudes for this double-barrier potential are expressed in terms of transmission and reflection amplitudes for its subparts. As the length of the constant-potential region can be arbitrary and the subparts of a potential may, in turn, be arbitrary segmented potentials, one obtains formulas recurerence formulas which express the scattering amplitudes for the arbitrary segmented potential via the scattering amplitudes for the subparts into which the complete potential can be divided. The efficiency of the method is demonstrated by solving analytically the problem of scattering from locally periodic potentials. Since an arbitrary potential can be approximated by a set of infinitely narrow rectangular barriers, the recurrence formulas can be applied to any potential, giving, in the limit of zero-width segments, differential equations for transmission, and reflection amplitudes.

Richard A. Crane

Noriaki Horiuchi

Kurt Schönhammer

The Anderson orthogonality exponent K for the overlap of two ground states of a free-electron gas with a local potential at different positions is discussed. In the small-distance limit, it is shown that for arbitrary local potentials K can be expressed in terms of the friction tensor. In the one-dimensional case, K depends on the potential only via the friction coefficient for arbitrary distance, and the exponent is bounded for arbitrary shape of the potential in contrast to the two- or higher-dimensional case. The exact result for K is presented for a short-range potential on a lattice.

A. A. Kozhevnikov

Temperature of the Bose -- Einstein condensation and the temperature behavior of the chemical potential and other thermodynamical functions of the ideal Bose gas are found for the arbitrary power-like spherical-symmetric potential at an arbitrary space dimension. It is shown that the recently observed Bose -- Einstein condensation of photons in the cavity is the phase transition of the third kind.