Semiclassical Orthogonal Polynomials, Matrix Models and Isomonodromic Tau Functions

The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such measures. These are shown to preserve the generalized monodromy of the associated rank-2 rational covariant derivative operators. The corresponding matrix models, consisting of unitarily diagonalizable matrices with spectra supported on these contours are analyzed, and it is shown that all coefficients of the associated spectral curves are given by logarithmic derivatives of the partition function or, more generally, the gap probabilities. The associated isomonodromic tau functions are shown to coincide, within an explicitly computed factor, with these partition functions. Research supported in part by the Natural Sciences and Engineering Research Council of Canada, the Fonds FCAR du Québec and EC ITH Network HPRN-CT-1999-000161.     

Fermionic construction of the partition function for multimatrix models and the multicomponent Toda lattice hierarchy

We use p-component fermions, p = 2, 3,..., to represent (2p−2)N-fold integrals as a fermionic vacuum expectation. This yields a fermionic representation for various (2p−2)-matrix models. We discuss links with the p-component Kadomtsev-Petviashvili hierarchy and also with the p-component Toda lattice hierarchy. We show that the set of all but two flows of the p-component Toda lattice hierarchy changes standard matrix models to new ones. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 265–277, August, 2007.     

Quadratic pseudopotentials for Gl(N, ) principal sigma models

Matrix pseudopotentials for the Gl(n, ) principal sigma model in two-dimensional Minkowski space are derived through the method of closed differential ideals. The structure of the pseudopotential equations is chosen to contain only quadratic nonlinearities of the matrix Riccati type. A set of further constraints is deduced which leads to inner Bäcklund transformations depending on two complex parameters. Through a geometrical interpretation of such matrix Riccati equations and the associated constraints, a linearization procedure is derived and the equations are reduced to those of Zakharov, Mikhaïlov and Shabat [7–10]. An inductive procedure is applied to explicitly solve an iterated sequence of such Bäcklund transformations through a purely algebraic transformation of solutions to the first step. The resulting nonlinear superposition formula is used to demonstrate a permutability theorem, and to analyse the appearance of singularities.fr|Nous déduisons les pseudo-potentiels matriciels pour le modèle sigma principal à valeurs dans Gl(N, ), défini sur l'espace de Minkowski bi-dimensionnel, en utilisant la méthode des idéaux différentiels fermés. On restreint la structure des équations des pseudopotentiels de façon à ce que n'apparaissent que des non-linéarités quadratiques du type Riccati matriciel. Nous déduisons un ensemble de contraintes supplémentaires menant à des transformations de Bäcklund internes dépendantes de deux paramètres complexes. Grâce à l'interprétation géométrique de ces équations de Riccati matricielles et des contraintes associées, nous déduisons un procédé de linéarisation et les équations se réduisent à celles de Zakharov, Mikhailov et Shabat [7–10]. Un procédé inductif est utilisé pour résoudre explicitement une succession de transformations de Bäcklund par une transformation purement algébrique des solutions de la première ètape. La formule de superposition non-linéaire qui en résulte permet de démontrer un théorème de permutabilité et d'analyser l'apparition de singularités.     

The eikonal approximation and E(2) invariance

A group theoretic interpretation is given for the eikonal approximation in potential scattering. This is based upon the approximate invariance at high energies under translations and rotations in the transverse scattering plane; that is, symmetry under the group E(2). The Lippman-Schwinger equation is formulated in a set of basis states which transform invariantly under irreducible representations of E(2) and the solution for the eikonal Hamiltonian, together with lowest order (Saxon-Schiff) corrections is obtained within this basis. A formulation of unitarity in the impact-parameter representation, based upon the E(2) invariance is given. The “geometrical” interpretation of this representation, in connection with the eikonal approximation, is made clear by our approach.     

Generalized Drinfeld-Sokolov reductions and KdV type hierarchies

Generalized Drinfeld-Sokolov (DS) hierarchies are constructed through local reductions of Hamiltonian flows generated by monodromy invariants on the dual of a loop algebra. Following earlier work of De Groot et al., reductions based upon graded regular elements of arbitrary Heisenberg subalgebras are considered. We show that, in the case of the nontwisted loop algebra (gl _n ), graded regular elements exist only in those Heisenberg subalgebras which correspond either to the partitions ofn into the sum of equal numbersn=pr or to equal numbers plus onen=pr+1. We prove that the reduction belonging to the grade 1 regular elements in the casen=pr yields thep×p matrix version of the Gelfand-Dickeyr-KdV hierarchy, generalizing the scalar casep=1 considered by DS. The methods of DS are utilized throughout the analysis, but formulating the reduction entirely within the Hamiltonian framework provided by the classical r-matrix approach leads to some simplifications even forp=1.     

Darboux coordinates and Liouville-Arnold integration in loop algebras

Darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part of loop algebras. These are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves, defined within a given matrix representation. A Liouville generating function is obtained in completely separated form and shown, through the Liouvile-Arnold integration method, to lead to the Abel map linearization of all Hamiltonian flows induced by the spectral invariants. As illustrative examples, the caseg =sl(2), together with its real forms, is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics, with the Liouville generating function expressed in hyperellipsoidal coordinates. Forg =sl(3), the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schrödinger equation, a case which requires further symplectic constraints in order to deal with singularities in the spectral data at .Research supported in part by the Natural Sciences and Engineering Research Council of Canada and by U.S. Army Grant DAA L03-87-K-0110     

Superharmonic Perturbations of a Gaussian Measure, Equilibrium Measures and Orthogonal Polynomials

This work concerns superharmonic perturbations of a Gaussian measure given by a special class of positive weights in the complex plane of the form w(z) = exp(−|z|² + U^μ(z)), where U^μ(z) is the logarithmic potential of a compactly supported positive measure μ. The equilibrium measure of the corresponding weighted energy problem is shown to be supported on subharmonic generalized quadrature domains for a large class of perturbing potentials U^μ(z). It is also shown that the 2 × 2 matrix d-bar problem for orthogonal polynomials with respect to such weights is well-defined and has a unique solution given explicitly by Cauchy transforms. Numerical evidence is presented supporting a conjectured relation between the asymptotic distribution of the zeroes of the orthogonal polynomials in a semi-classical scaling limit and the Schwarz function of the curve bounding the support of the equilibrium measure, extending the previously studied case of harmonic polynomial perturbations with weights w(z) supported on a compact domain. Submitted: July 25, 2008. Accepted: October 1, 2008. Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds de recherche sur la nature et les technologies du Québec (FQRNT).     

Integrable Fredholm Operators and Dual Isomonodromic Deformations

The Fredholm determinants of a special class of integrable integral operators K supported on the union of m curve segments in the complex λ-plane are shown to be the τ-functions of an isomonodromic family of meromorphic covariant derivative operators , having regular singular points at the 2m endpoints of the curve segments, and a singular point of Poincaré index 1 at infinity. The rank r of the corresponding vector bundle over the Riemann sphere equals the number of distinct terms in the exponential sum defining the numerator of the integral kernel. The matrix Riemann–Hilbert problem method is used to deduce an identification of the Fredholm determinant as a τ-function in the sense of Segal–Wilson and Sato, i.e., in terms of abelian group actions on the determinant line bundle over a loop space Grassmannian. An associated dual isomonodromic family of covariant derivative operators , having rank n= 2m, and r finite regular singular points located at the values of the exponents defining the kernel of K is derived. The deformation equations for this family are shown to follow from an associated dual set of Riemann–Hilbert data, in which the r?les of the r exponential factors in the kernel and the 2m endpoints of its support are interchanged. The operators are analogously associated to an integral operator whose Fredholm determinant is equal to that of K. Received: 10 June 1997 / Received revised: 16 February 2001 / Accepted: 27 November 2001     

The soliton correlation matrix and the reduction problem for integrable systems

Integrable 1+1 dimensional systems associated to linear first-order matrix equations meromorphic in a complex parameter, as formulated by Zakharov, Mikhailov, and Shabat [1–3] (ZMS) are analyzed by a new method based upon the soliton correlation matrix (M-matrix). The multi-Bäcklund transformation, which is equivalent to the introduction of an arbitrary number of poles in the ZMS dressing matrix, is expressed by a pair of matrix Riccati equations for theM-matrix. Through a geometrical interpretation based upon group actions on Grassman manifolds, the solution of this system is explicitly determined in terms of the solutions to the ZMS linear system. Reductions of the system corresponding to invariance under finite groups of automorphisms are also solved by reducing theM-matrix suitably so as to preserve the class of invariant solutions.Supported in part by the Natural Sciences and Engineering Research Council of Canada and the Fonds FCAC pour l'aide et le soutien à la recherche     

Duality of Spectral Curves Arising in Two-Matrix Models

We consider the two-matrix model with the measure given by the exponential of a sum of polynomials in two different variables. We derive a sequence of pairs of dual finite-size systems of ODEs for the corresponding biorthonormal polynomials. We prove an inverse theorem, which shows how to reconstruct such measures from pairs of semi-infinite finite-band matrices, which define the recursion relations and satisfy the string equation. In the limit N, we prove that the obtained dual systems have the same spectral curve.