The phase diagram of the multi-dimensional Anderson localization via analytic determination of Lyapunov exponents

The method proposed by the present authors to deal analytically with the problem of Anderson localization via disorder [J.Phys.: Condens. Matter {\bf 14} (2002) 13777] is generalized for higher spatial dimensions D. In this way the generalized Lyapunov exponents for diagonal correlators of the wave function, $$, can be calculated analytically and exactly. This permits to determine the phase diagram of the system. For all dimensions $D > 2$ one finds intervals in the energy and the disorder where extended and localized states coexist: the metal-insulator transition should thus be interpreted as a first-order transition. The qualitative differences permit to group the systems into two classes: low-dimensional systems ($2\leq D \leq 3$), where localized states are always exponentially localized and high-dimensional systems ($D\geq D_c=4$), where states with non-exponential localization are also formed. The value of the upper critical dimension is found to be $D_0=6$ for the Anderson localization problem; this value is also characteristic of a related problem - percolation.Comment: 17 pages, 5 figures, to appear in Eur. Phys.J.

Reply to Comment on "Exact analytic solution for the generalized Lyapunov exponent of the 2-dimensional Anderson localization"

We reply to comments by P.Marko$\breve{s}$, L.Schweitzer and M.Weyrauch [preceding paper] on our recent paper [J. Phys.: Condens. Matter 63, 13777 (2002)]. We demonstrate that our quite different viewpoints stem for the different physical assumptions made prior to the choice of the mathematical formalism. The authors of the Comment expect \emph{a priori} to see a single thermodynamic phase while our approach is capable of detecting co-existence of distinct pure phases. The limitations of the transfer matrix techniques for the multi-dimensional Anderson localization problem are discussed.Comment: 4 pages, accepted for publication in J.Phys.: Condens. Mat

Применение треугольной схемы расположения скважин для добычи урана из малого изометрического рудного тела

The dynamical behaviour of the laser beam fusion cutting process of metals is investigated. Integral methods such as the variational formulation are applied to the partial differential equations for the free boundary problem and a finite dimensional approximation of the dynamical system is obtained. The model describes the shape of the evolving cutting kerf and the melt flow. The analysis is aimed at revealing the characteristic features of the resultant cut, for example, ripple formation and adherent dross. The formation of the ripples in the upper part of the put, where no resolidified material is detectable, is discussed in detail. A comparison with numerical simulations and experiments is made