Exact Solution to the Moment Problem for the XY Chain

We present the exact solution to the moment problem for the spin-1/2 isotropic antiferromagnetic XY chain with explicit forms for the moments with respect to the Neel state, the cumulant generating function, and the Resolvent Operator. We verify the correctness of the Horn-Weinstein Theorems, but the analytic structure of the generating function in the complex t-plane is quite different from that assumed by the "t"-Expansion and the Connected Moments Expansion due to the vanishing gap. This function has a finite radius of convergence about t=0, and for large t has a leading descending algebraic series E(t)-E_0 ~ At^{-2}. The Resolvent has a branch cut and essential singularity near the ground state energy of the form G(s)/s ~ B|s+1|^{-3/4} exp(C|s+1|^{1/2}). Consequently extrapolation strategies based on these assumptions are flawed and in practise we find that the CMX methods are pathological and cannot be applied, while numerical evidence for two of the "t"-expansion methods indicates a clear asymptotic convergence behaviour with truncation order.Comment: 15 pages + 2 postscript files, Latex2e + amstex + amssyb, to appear in Int. J. Mod. Phys.

Semi-classical Orthogonal Polynomial Systems on Non-uniform Lattices, Deformations of the Askey Table and Analogs of Isomonodromy

A $\mathbb{D}$-semi-classical weight is one which satisfies a particular linear, first order homogeneous equation in a divided-difference operator $\mathbb{D}$. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first order homogeneous matrix equation in the divided-difference operator termed the spectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the $\mathbb{D}$-semi-classical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the $\mathbb{D}$-semi-classical class it is entirely natural to define a generalisation of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first non-trivial deformation of the Askey-Wilson orthogonal polynomial system defined by the $q$-quadratic divided-difference operator, the Askey-Wilson operator, and derive the coupled first order divided-difference equations characterising its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to the $E^{(1)}_7$ $q$-Painlev\'e system.Comment: Submitted to Duke Mathematical Journal on 5th April 201

Construction of a Lax Pair for the E6(1)E_6^{(1)} qq-Painlev\'e System

We construct a Lax pair for the $E^{(1)}_6$ $q$-Painlev\'e system from first principles by employing the general theory of semi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic lattices [arXiv:1204.2328]. Our study treats one special case of such lattices - the $q$-linear lattice - through a natural generalisation of the big $q$-Jacobi weight. As a by-product of our construction we derive the coupled first-order $q$-difference equations for the $E^{(1)}_6$ $q$-Painlev\'e system, thus verifying our identification. Finally we establish the correspondences of our result with the Lax pairs given earlier and separately by Sakai and Yamada, through explicit transformations