Fractional Derivative as Fractional Power of Derivative

Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of self-adjoint derivative operator. The Fourier integrals and Weyl quantization procedure are applied to derive the definition of fractional derivative operator. Fractional generalization of concept of stability is considered.Comment: 20 pages, LaTe

"Unusual" metals in two dimensions: one-particle model of the metal-insulator transition at T=0

The conductance of disordered nano-wires at T=0 is calculated in one-particle approximation by reducing the original multi-dimensional problem for an open bounded system to a set of exactly one-dimensional non-Hermitian problems for mode propagators. Regarding two-dimensional conductor as a limiting case of three-dimensional disordered quantum waveguide, the metallic ground state is shown to result from its multi-modeness. On thinning the waveguide (in practice, e. g., by means of the ``pressing'' external electric field) the electron system undergoes a continuous phase transition from metallic to insulating state. The result predicted conform qualitatively to the observed anomalies of the resistance of different planar electron and hole systems.Comment: 7 pages, LATEX-2

Highest coefficient of scalar products in SU(3)-invariant integrable models

We study SU(3)-invariant integrable models solvable by nested algebraic Bethe ansatz. Scalar products of Bethe vectors in such models can be expressed in terms of a bilinear combination of their highest coefficients. We obtain various different representations for the highest coefficient in terms of sums over partitions. We also obtain multiple integral representations for the highest coefficient.Comment: 17 page

Metal-insulator transition in a two-dimensional electron system: the orbital effect of in-plane magnetic field

The conductance of an open quench-disordered two-dimensional (2D) electron system subject to an in-plane magnetic field is calculated within the framework of conventional Fermi liquid theory applied to actually a three-dimensional system of spinless electrons confined to a highly anisotropic (planar) near-surface potential well. Using the calculation method suggested in this paper, the magnetic field piercing a finite range of infinitely long system of carriers is treated as introducing the additional highly non-local scatterer which separates the circuit thus modelled into three parts -- the system as such and two perfect leads. The transverse quantization spectrum of the inner part of the electron waveguide thus constructed can be effectively tuned by means of the magnetic field, even though the least transverse dimension of the waveguide is small compared to the magnetic length. The initially finite (metallic) value of the conductance, which is attributed to the existence of extended modes of the transverse quantization, decreases rapidly as the magnetic field grows. This decrease is due to the mode number reduction effect produced by the magnetic field. The closing of the last current-carrying mode, which is slightly sensitive to the disorder level, is suggested as the origin of the magnetic-field-driven metal-to-insulator transition widely observed in 2D systems.Comment: 19 pages, 7 eps figures, the extension of cond-mat/040613

On the continuum limit for discrete NLS with long-range lattice interactions

We consider a general class of discrete nonlinear Schroedinger equations (DNLS) on the lattice $h \mathbb{Z}$ with mesh size $h>0$. In the continuum limit when $h \to 0$, we prove that the limiting dynamics are given by a nonlinear Schroedinger equation (NLS) on $\mathbb{R}$ with the fractional Laplacian $(-\Delta)^\alpha$ as dispersive symbol. In particular, we obtain that fractional powers $1/2 < \alpha < 1$ arise from long-range lattice interactions when passing to the continuum limit, whereas NLS with the non-fractional Laplacian $-\Delta$ describes the dispersion in the continuum limit for short-range lattice interactions (e.g., nearest-neighbor interactions). Our results rigorously justify certain NLS model equations with fractional Laplacians proposed in the physics literature. Moreover, the arguments given in our paper can be also applied to discuss the continuum limit for other lattice systems with long-range interactions.Comment: 26 pages; no figures. Some minor revisions. To appear in Comm. Math. Phy

Three realizations of quantum affine algebra Uq(A2(2))U_q(A_2^{(2)})

In this article we establish explicit isomorphisms between three realizations of quantum twisted affine algebra $U_q(A_2^{(2)})$: the Drinfeld ("current") realization, the Chevalley realization and the so-called $RLL$ realization, investigated by Faddeev, Reshetikhin and Takhtajan.Comment: 15 page