Term Recursion Relations Research Articles

Macdonald Polynomials of Type C<sub>n</sub> with One-Column Diagrams and Deformed Catalan Numbers

We present an explicit formula for the transition matrix $\mathcal{C}$ from the type $C_n$ degeneration of the Koornwinder polynomials $P_{(1^r)}(x\,|\,a,-a,c,-c\,|\,q,t)$ with one column diagrams, to the type $C_n$ monomial symmetric polynomials $m_{(1^{r})}(x)$. The entries of the matrix $\mathcal{C}$ enjoy a set of three term recursion relations, which can be regarded as a $(a,c,t)$-deformation of the one for the Catalan triangle or ballot numbers. Some transition matrices are studied associated with the type $(C_n,C_n)$ Macdonald polynomials $P^{(C_n,C_n)}_{(1^r)}(x\,|\,b;q,t)= P_{(1^r)}\big(x\,|\,b^{1/2},-b^{1/2},q^{1/2}b^{1/2},-q^{1/2}b^{1/2}\,|\,q,t\big)$. It is also shown that the $q$-ballot numbers appear as the Kostka polynomials, namely in the transition matrix from the Schur polynomials $P^{(C_n,C_n)}_{(1^r)}(x\,|\,q;q,q)$ to the Hall-Littlewood polynomials $P^{(C_n,C_n)}_{(1^r)}(x\,|\,t;0,t)$.

Pre-sequences of matrix orthogonal polynomials

We introduce the notion of a pre-sequence of matrix orthogonal polynomials to mean a sequence $$\{F_n\}_{n\ge 0}$$ of matrix orthogonal functions with respect to a weight function W, satisfying a three term recursion relation and such that $$\det (F_{0})$$ is not zero almost everywhere. By now there is a uniform construction of such sequences from irreducible spherical functions of some fixed K-types associated to compact symmetric pairs (G, K) of rank one. Our main result is that $$\{Q_n=F_nF_0^{-1}\}_{n\ge 0}$$ is a sequence of matrix orthogonal polynomials with respect to the weight function $$F_0WF_0^*$$ , see Theorem 21.

Reducibility of matrix weights

In this paper we discuss the notion of reducibility for matrix weights and introduce a real vector space $\mathcal C_\mathbb{R}$ which encodes all information about the reducibility of $W$. In particular a weight $W$ reduces if and only if there is a non-scalar matrix $T$ such that $TW=WT^*$. Also, we prove that reducibility can be studied by looking at the commutant of the monic orthogonal polynomials or by looking at the coefficients of the corresponding three term recursion relation. A matrix weight may not be expressible as direct sum of irreducible weights, but it is always equivalent to a direct sum of irreducible weights. We also establish that the decompositions of two equivalent weights as sums of irreducible weights have the same number of terms and that, up to a permutation, they are equivalent. We consider the algebra of right-hand-side matrix differential operators $\mathcal D(W)$ of a reducible weight $W$, giving its general structure. Finally, we make a change of emphasis by considering reducibility of polynomials, instead of reducibility of matrix weights.

NONPLANAR INTEGRABILITY: BEYOND THE SU(2) SECTOR

We compute the one-loop anomalous dimensions of restricted Schur polynomials with a classical dimension Δ~O(N). The operators that we consider are labeled by Young diagrams with two long columns or two long rows. Simple analytic expressions for the action of the dilatation operator are found. The projection operators needed to define the restricted Schur polynomials are constructed by translating the problem into a spin chain language, generalizing earlier results obtained in the SU(2) sector of the theory. The diagonalization of the dilatation operator reduces to solving five term recursion relations. The recursion relations can be solved exactly in terms of products of symmetric Kravchuk polynomials with Hahn polynomials. This proves that the dilatation operator reduces to a decoupled set of harmonic oscillators and therefore it is integrable, extending a similar conclusion reached for the SU(2) sector of the theory.

Nonplanar integrability

In this article we study operators with a dimension $\Delta\sim O(N)$ and show that simple analytic expressions for the action of the dilatation operator can be found. The operators we consider are restricted Schur polynomials. There are two distinct classes of operators that we consider: operators labeled by Young diagrams with two long columns or two long rows. The main complication in working with restricted Schur polynomials is in building a projector from a given $S_{n+m}$ irreducible representation to an $S_n\times S_m$ irreducible representation (both specified by the labels of the restricted Schur polynomial). We give an explicit construction of these projectors by reducing it to the simple problem of addition of angular momentum in ordinary non-relativistic quantum mechanics. The diagonalizationof the dilatation operator reduces to solving three term recursion relations. The fact that the recursion relations have only three terms is a direct consequence of the weak mixing at one loop of the restricted Schur polynomials. The recursion relations can be solved exactly in terms of symmetric Kravchuk polynomials or in terms of Clebsch-Gordan coefficients. This proves that the dilatation operator reduces to a decoupled set of harmonic oscillators and therefore it is integrable.

Bulk asymptotics of skew-orthogonal polynomials for quartic double well potential and universality in the matrix model

We derive bulk asymptotics of skew-orthogonal polynomials πm(β), β=1, 4, defined with respect to the weight exp(−2NV(x)), V(x)=gx4/4+tx2/2, g>0 and t<0. We assume that as m,N→∞, there exists an ϵ>0 such that ϵ≤(m/N)≤λcr−ϵ, where λcr is the critical value that separates skew-orthogonal polynomials with two cuts from those with one cut. Simultaneously we derive asymptotics for the recursive coefficients of skew-orthogonal polynomials. The proof is based on obtaining a finite term recursion relation between skew-orthogonal polynomials and orthogonal polynomials and using asymptotic results of orthogonal polynomials derived by Bleher and Its [“Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problems, and universality in the matrix model,” Ann. Math. 150, 185 (1999)]. Finally, we apply these asymptotic results of skew-orthogonal polynomials and their recursion coefficients in the generalized Christoffel–Darboux formula [Ghosh, S., “Generalized Christoffel-Darboux formula for skew-orthogonal polynomials and random matrix theory,” J. Phys. A 39, 8775 (2006)] to obtain level densities and sine kernels in the bulk of the spectrum for orthogonal and symplectic ensembles of random matrices.

Exact Heisenberg operator solutions for multiparticle quantum mechanics

Exact Heisenberg operator solutions for independent “sinusoidal coordinates” as many as the degree of freedom are derived for typical exactly solvable multiparticle quantum mechanical systems, the Calogero systems [J. Math. Phys. 12, 419 (1971)] based on any root system. These Heisenberg operator solutions also present the explicit forms of the annihilation-creation operators for various quanta in the interacting multiparticle systems. At the same time they can be interpreted as multivariable generalization of the three term recurrence relations for multivariable orthogonal polynomials constituting the eigenfunctions.

Three Term Recursion Relation for Spherical Functions Associated to the Complex Projective Plane

The aim of this paper is to prove a three term recursion relation for a sequence of matrix valued functions \(\widetilde\Phi\)(g,w) on G=SU(3) built up of l+1 spherical functions of a given type πn,l, associated to the complex projective plane G/K, K=S(U(2)×U(1)). The three term recursion relation that constitutes our main result, Theorem 5.2, together with the fact that the functions \(\widetilde\Phi\)(g,w) are eigenfunctions of all differential operators on G which are left invariant under G and right invariant under K, provides for each l∈N0 a solution of a matrix valued extension of the Bochner's problem to G. In fact by restriction to an Abelian Iwasawa subgroup of G, for each l∈N0, we obtain a sequence \({\widetilde H}\)(t,w) of matrix valued polynomial functions on t which satisfies a three term recursion relation and such that they are eigenfunctions of a second order differential operator on 02<t<1. Thus each sequence \({\widetilde H}\)(t,w) satisfies both conditions explicitly asked for by Bochner.

A polynomial expansion of the quantum propagator, the Green’s function, and the spectral density operator

One of the methods for calculating time propagators in quantum mechanics uses an expansion of e−iĤt/ℏ in a sum of orthogonal polynomial. Equations involving Chebychev, Legendre, Laguerre, and Hermite polynomials have been used so far. We propose a new formula, in which the propagator is expressed as a sum in which each term is a Gegenbauer polynomial multiplied with a Bessel function. The equations used in previous work can be obtained from ours by giving specific values to a parameter. The expression allows analytic continuation from imaginary to real time, transforming thus results obtained by evaluating thermal averages into results pertaining to the time evolution of the system. Starting from the expression for the time propagator we derive equations for the Green’s function and the density of states. To perform computations one needs to calculate how the polynomial in the Hamiltonian operator acts on a wave function. The high order polynomials can be obtained from the lower ordered ones through a three term recursion relation; this saves storage and computer time. As a numerical test, we have computed the bound state spectrum of the Morse oscillator and the transmission coefficient for tunneling through an Eckart barrier. We have also studied the evolution of a Gaussian wave packet in a Morse potential well.

Electrostatic interpretation for the zeros of certain polynomials and the Darboux process

The purpose of this paper is to extend the electrostatic interpretation of Stieltjes to the zeros of a larger class of important polynomials. To keep things simple I consider polynomials that are obtained from those considered by Stieltjes by means of the so-called Darboux process applied to the difference operator featuring in their three term recursion relation. It will turn out that some of the beauty in the results of Stieltjes is preserved in this process. It would be interesting to interpret the Darboux process at the electrostatic level and this paper gives “experimental data” that might help in this direction.

Bispectral darboux transformations: An extension of the Krall polynomials

Orthogonal polynomials satisfying fourth order differential equations were classified by H. L. Krall [K2]. They can be obtained from very special instances of the (generalized) Laguerre and the Jacobi polynomials by the Darboux process applied to semi-infinite tridiagonal matrices [GH1]. In this paper, starting from any instance of the (generalized) Laguerre and the Jacobi polynomials, we construct a one parameter family of polynomials which are eigenfunctions of a fourth order differential operator. In general, these polynomials will satisfy a five term recursion relation. Some further examples involving higher order differential operators and recursion relations are also presented. We use a proper version of the idea of a bispectral involution first proposed by G. Wilson [W1] and then reinterpreted in the context of the Darboux transformation by other authors [KR], [BHY1]. The notion of a second order bispectral differential operator was introduced in [DG]. One says that such an operator L = −d2/dx2 + V(x) is bispectral if there exists a family of its eigenfunctions L(x, d/dx)ψ(x, k) = kψ(x, k)

Orthogonality properties of iterated polynomial mappings

We consider a measure defined on a complex contour and its associated orthogonal polynomials. The action of a polynomial transformation on the measure and the transformation laws of the corresponding orthogonal polynomials are given. Iterating the transformation provides an invariant measure, whose support is the Julia set corresponding to the polynomial transformation. It appears that, up to a constant, the iterated polynomials generated by the initial mapping form a subset of the invariant set of orthogonal polynomials, which fulfill a three term recursion relation. An algorithm is given to compute the coefficients of this recursion relation, which can be interpreted as a linear extension of the iterative procedure.

The occupational statistics for indistinguishable trimers on a 3×N lattice space

A 20 term recursion relation is derived that describes exactly the occupational degeneracy for trimers distributed on a 3×N lattice space. The associated generating functions, expectation, normalization, dispersion, and continuous representation are also developed and discussed.

On the functions associated with the Burrau approximation for the hydrogen molecular ion and analogous molecules

The Schrödinger equation for the Burrau-like molecular species (A-B)ZA+ZB−1 is (−h/2/2me) ∇2 ψ + [−ZAe2/rA−ZBe2/rB +ZAZBe2/RAB]ψ = Eψ. After separation using prolate spheroidal coordinates, the mathematical analysis of this equation can be reduced to the investigation of the ordinary differential equation (d/dx) { (1−x2)[dP (x)/dx]} + {[−m2/(1−x2)] + 2αβx + α2x2 + σ} P (x) =0, where m is an integer, α and β are molecular parameters expressed as elements of the real number set, and σ is the eigenvalue of the equation which is dependent on two numbers κ and m related to the quantum numbers found in the discussion of the hydrogen atom as well as α and β. The solutions for the function P (x) where m is replaced by μ and μ is any number are expanded in power series of various kinds, in series of Legendre functions, and confluent hypergeometric functions to give three term recursion relations. The recursion relations are used to generate continued fractions which allow convenient numerical calculation of the eigenvalues and they are also helpful in the derivation of asymptotic expressions of the eigenvalues and eigenfunctions in terms of their dependent variables. Integral equation solutions are given for the two intervals x∈[−1,1] and x∈[1,∞]. For β=−(ν+μ+1) or β=ν+μ+1, where ν∈{0} UN, it is possible to generate polynomials solutions.