Integrable superconductivity and Richardson equations

For the integrable generalized model of superconductivity a solution of the Richardson equations for a spectrum of model is studied. For the case of narrow band the solution is presented in terms of the generalized Laguerre or Jacobi polynomials. In asymptotic limit, when the Richardson equations are transformed to an integral singular equation, the properties of an integration contour are discussed and a spectral density is calculated. Conditions for appearance of gaps in the spectrum are considered.Для інтегровної узагальненої моделі надпровідності досліджено розв'язання рівнянь Річардсона для спектра моделі. У випадку вузької зони розв'язок подано в термінах узагальнених поліномів Лагерра та Якобі. В асимптотичному випадку, коли рівняння Річардсона трансформуються в інтегральне сингулярне рівняння, з'ясовано властивості контура інтегрування та розраховано спектральну щільність. Розглянуто умови появи щілин у спектрі

Mendeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential

We prove that a neutral atom in mean-field approximation has O(4) symmetry and this fact explains the empirical [n+l,n]-rule or Madelung rule which describes effectively periods, structure and other properties of the Mendeleev table of chemical elements

Superconductivity and integrability

The paper is a review of studies of integrability of the BCS Hamiltonian with discussion of some its integrable generalization which present an interest for a number of physical problems

Lam\'e polynomials, hyperelliptic reductions and Lam\'e band structure

The band structure of the Lam\'e equation, viewed as a one-dimensional Schr\"odinger equation with a periodic potential, is studied. At integer values of the degree parameter l, the dispersion relation is reduced to the l=1 dispersion relation, and a previously published l=2 dispersion relation is shown to be partially incorrect. The Hermite-Krichever Ansatz, which expresses Lam\'e equation solutions in terms of l=1 solutions, is the chief tool. It is based on a projection from a genus-l hyperelliptic curve, which parametrizes solutions, to an elliptic curve. A general formula for this covering is derived, and is used to reduce certain hyperelliptic integrals to elliptic ones. Degeneracies between band edges, which can occur if the Lam\'e equation parameters take complex values, are investigated. If the Lam\'e equation is viewed as a differential equation on an elliptic curve, a formula is conjectured for the number of points in elliptic moduli space (elliptic curve parameter space) at which degeneracies occur. Tables of spectral polynomials and Lam\'e polynomials, i.e., band edge solutions, are given. A table in the older literature is corrected.Comment: 38 pages, 1 figure; final revision

On the calculation of finite-gap solutions of the KdV equation

A simple and general approach for calculating the elliptic finite-gap solutions of the Korteweg-de Vries (KdV) equation is proposed. Our approach is based on the use of the finite-gap equations and the general representation of these solutions in the form of rational functions of the elliptic Weierstrass function. The calculation of initial elliptic finite-gap solutions is reduced to the solution of the finite-band equations with respect to the parameters of the representation. The time evolution of these solutions is described via the dynamic equations of their poles, integrated with the help of the finite-gap equations. The proposed approach is applied by calculating the elliptic 1-, 2- and 3-gap solutions of the KdV equations

A Symplectic Structure for String Theory on Integrable Backgrounds

We define regularised Poisson brackets for the monodromy matrix of classical string theory on R x S^3. The ambiguities associated with Non-Ultra Locality are resolved using the symmetrisation prescription of Maillet. The resulting brackets lead to an infinite tower of Poisson-commuting conserved charges as expected in an integrable system. The brackets are also used to obtain the correct symplectic structure on the moduli space of finite-gap solutions and to define the corresponding action-angle variables. The canonically-normalised action variables are the filling fractions associated with each cut in the finite-gap construction. Our results are relevant for the leading-order semiclassical quantisation of string theory on AdS_5 x S^5 and lead to integer-valued filling fractions in this context.Comment: 41 pages, 2 figures; added references, corrected typos, improved discussion of Hamiltonian constraint

Noncommutative geometry, Quantum effects and DBI-scaling in the collapse of D0-D2 bound states

We study fluctuations of time-dependent fuzzy two-sphere solutions of the non-abelian DBI action of D0-branes, describing a bound state of a spherical D2-brane with N D0-branes. The quadratic action for small fluctuations is shown to be identical to that obtained from the dual abelian D2-brane DBI action, using the non-commutative geometry of the fuzzy two-sphere. For some of the fields, the linearized equations take the form of solvable Lam\'e equations. We define a large-N DBI-scaling limit, with vanishing string coupling and string length, and where the gauge theory coupling remains finite. In this limit, the non-linearities of the DBI action survive in both the classical and the quantum context, while massive open string modes and closed strings decouple. We describe a critical radius where strong gauge coupling effects become important. The size of the bound quantum ground state of multiple D0-branes makes an intriguing appearance as the radius of the fuzzy sphere, where the maximal angular momentum quanta become strongly coupled.Comment: 34 pages, Latex; v2: Minor correction in conformal transformation of couplings, references adde

Closed geodesics and billiards on quadrics related to elliptic KdV solutions

We consider algebraic geometrical properties of the integrable billiard on a quadric Q with elastic impacts along another quadric confocal to Q. These properties are in sharp contrast with those of the ellipsoidal Birkhoff billiards. Namely, generic complex invariant manifolds are not Abelian varieties, and the billiard map is no more algebraic. A Poncelet-like theorem for such system is known. We give explicit sufficient conditions both for closed geodesics and periodic billiard orbits on Q and discuss their relation with the elliptic KdV solutions and elliptic Calogero systemComment: 23 pages, Latex, 1 figure Postscrip

Hyperelliptic Theta-Functions and Spectral Methods: KdV and KP solutions

This is the second in a series of papers on the numerical treatment of hyperelliptic theta-functions with spectral methods. A code for the numerical evaluation of solutions to the Ernst equation on hyperelliptic surfaces of genus 2 is extended to arbitrary genus and general position of the branch points. The use of spectral approximations allows for an efficient calculation of all characteristic quantities of the Riemann surface with high precision even in almost degenerate situations as in the solitonic limit where the branch points coincide pairwise. As an example we consider hyperelliptic solutions to the Kadomtsev-Petviashvili and the Korteweg-de Vries equation. Tests of the numerics using identities for periods on the Riemann surface and the differential equations are performed. It is shown that an accuracy of the order of machine precision can be achieved.Comment: 16 pages, 8 figure

Giant Magnons and Singular Curves

We obtain the giant magnon of Hofman-Maldacena and its dyonic generalisation on R x S^3 < AdS_5 x S^5 from the general elliptic finite-gap solution by degenerating its elliptic spectral curve into a singular curve. This alternate description of giant magnons as finite-gap solutions associated to singular curves is related through a symplectic transformation to their already established description in terms of condensate cuts developed in hep-th/0606145.Comment: 34 pages, 17 figures, minor change in abstrac