Monodromy matrix

Abstract: Form factors for local spin operators of the XXZ Heisenberg spin-1/2 finite chain are computed. Representation theory of Drinfel'd twists for the sl2 quantum affine algebra in finite dimensional modules is used to calculate scalar products of Bethe states (leading to Gaudin formula) and to solve the quantum inverse problem for local spin operators in the finite XXZ chain. Hence, we obtain the representation of the n-spin correlation functions in terms of expectation values(in ferromagnetic reference state) of the operator entries of the quantum monodromy matrix satisfying Yang-Baxter algebra. This leads to the direct calculation of the form factors of the XXZ Heisenberg spin-1/2 finite chain as determinants of usual functions of the parameters of the model. A two-point correlation function for adjacent sites is also derived using similar techniques.

Abstract: A systematic study has been made of periodic orbits in the two-dimensional, elliptic, restricted three-body problem. All ranges of eccentricities, from 0 to 1, and mass-ratios, from 0 to J, have been investigated. Eleven hundred periodic orbits have been obtained. It is concluded that the elliptic problem behaves in a way which is completely different from the circular problem. The main difference is in the stability properties of the periodic orbits. Because of the nonexistence of the Jacobi integral (the elliptic problem is not conservative), the characteristic equation of the monodromy matrix does not have a pair of unit roots, in general. The stability is denned by two real numbers (stability indices) rather than one. Because of that, there are seven general classes of periodic orbits, according to their stability properties. The stability of the periodic orbits has been determined by numerically integrating the variational equations with a recurrent power series method. The results are in contrast with the circular problem, where there are only three classes of orbits (stability, even instability, and odd instability): in the elliptic problem there are one stable class and six unstable classes. The elliptic, restricted three-body problem can be considered as the prototype of all nonintegrable, nonconservative Hamiltonian systems, and in this paper, probably for the first time, a classification of the multipliers is given for these systems. I. Introduction

Abstract: In this paper we consider various regularity results for discrete quasiperiodic Schr6dinger equations --n+l - Pn-1 + V(9 + nw)on = EOn with analytic potential V. We prove that on intervals of positivity for the Lyapunov exponent the integrated density of states is Holder continuous in the energy provided w has a typical continued fraction expansion. The proof is based on certain sharp large deviation theorems for the norms of the monodromy matrices and the "avalanche-principle". The latter refers to a mechanism that allows us to write the norm of a monodromy matrix as the product of the norms of many short blocks. In the multi-frequency case the integrated density of states is shown to have a modulus of continuity of the form exp(- log tl) for some 0 < a < 1, but currently we do not obtain Holder continuity in the case of more than one frequency. We also present a mechanism for proving the positivity of the Lyapunov exponent for large disorders for a general class of equations. The only requirement for this approach is some weak form of a large deviation theorem for the Lyapunov exponents. In particular, we obtain an independent proof of the Herman-Sorets-Spencer theorem in the multi-frequency case. The approach in this paper is related to the recent nonperturbative proof of Anderson localization in the quasi-periodic case by J. Bourgain and M. Goldstein.