Stieltjes Polynomials Research Articles

Stieltjes analytic functions and higher order linear differential equations

In this work we develop a theory of Stieltjes-analytic functions. We first define the Stieltjes monomials and polynomials and we study them exhaustively. Then, we introduce the g-analytic functions locally, as an infinite series of these Stieltjes monomials and we study their properties in depth and how they relate to higher order Stieltjes differentiation. We define the exponential series and prove that it solves the first order linear problem. Finally, we apply the theory to solve higher order linear homogeneous Stieltjes differential equations with constant coefficients.

An exact solution of the homogenous trimer Bose-Hubbard model

It is shown that the homogenous three-site (trimer) Bose–Hubbard model with a periodic boundary condition can be solved exactly by using an extended Bethe ansatz based on the solution of the dimer model and the site-permutation symmetry. A solution of the model within the S 3 symmetric or non-symmetric subspace is presented. Coupled differential equations of a series of extended Heine–Stieltjes polynomials, the zeros of which are related to the solution of the model, are derived. Numerical examples of the solution of the model with bosons, including the related Heine–Stieltjes polynomials and the Van Vleck polynomials, are presented, which serve to validate the procedure and illustrate the completeness of the solutions they render.

Exact solution of the two-axis two-spin Hamiltonian

Bethe ansatz solution of the two-axis two-spin Hamiltonian is derived based on the Jordan–Schwinger boson realization of the SU(2) algebra. It is shown that the solution of the Bethe ansatz equations can be obtained as zeros of the related extended Heine–Stieltjes polynomials. Symmetry properties of excited levels of the system and zeros of the related extended Heine–Stieltjes polynomials are discussed. As an example of an application of the theory, the two equal spin case is studied in detail, which demonstrates that the levels in each band are symmetric with respect to the zero energy plane perpendicular to the level diagram and that excited states are always well entangled.

α − regular indefinite Stieltjes moment problem and Darboux transformation

A sequence of the real numbers $\textbf{s}=\{s_{i}\}_{i=0}^{\ell}$ is associated with the some indefinite Stieltjes moment problem and generalized Jacobi matrices. The relation between the $\alpha-$regular indefinite Stieltjes moment problem and shifted Darboux transformation of the generalized Jacobi matrix is studied. The new formulas for the Stieltjes polynomials with the shift are found and one are used to obtain the description of the solutions of the $\alpha-$regular indefinite Stieltjes moment problem. <br><br> Послідовність дійсних чисел $\textbf{s}=\{s_{i}\}_{i=0}^{\ell}$ пов'язана з деякою задачею про невизначений момент Стілтьєса та узагальненими матрицями Якобі. Досліджено зв'язок між $\alpha-$регулярною проблемою невизначеного моменту Стілтьєса та зміщеним перетворенням Дарбу узагальненої матриці Якобі. Знайдено нові формули для поліномів Стілтьєса зі зсувом та використано для отримання опису розв’язків $\alpha-$регулярної невизначеної проблеми моменту Стілтьєса.

Regularization of the indefinite Stieltjes moment problem

A not regular indefinite Stieltjes moment problem is studied in the generalized Stieltjes class Nκk. The new class of the sequence s={si}i=0ℓ is defined, which is called α-regular class. Regularization of the indefinite Stieltjes moment problem is obtained. The set of solutions of this problem is found in terms of the generalized Stieltjes polynomials and S-fractions with the shift α.

Addition Formula and Related Integral Equations for Heine—Stieltjes Polynomials

It is shown that symmetric products of Heine–Stieltjes quasi-polynomials satisfy an addition formula. The formula follows from the relationship between Heine–Stieltjes quasi-polynomials and spaces of generalized spherical harmonics, and from the known explicit form of the reproducing kernel of these spaces. In special cases, the addition formula is written out explicitly and verified. As an application, integral equations for Heine–Stieltjes quasi-polynomials are found.

An exact solution of spherical mean-field plus orbit-dependent non-separable pairing model with two non-degenerate j-orbits

An exact solution of nuclear spherical mean-field plus orbit-dependent non-separable pairing model with two non-degenerate j-orbits is presented. The extended one-variable Heine–Stieltjes polynomials associated to the Bethe ansatz equations of the solution are determined, of which the sets of the zeros give the solution of the model, and can be determined relatively easily. A comparison of the solution to that of the standard pairing interaction with constant interaction strength among pairs in any orbit is made. It is shown that the overlaps of eigenstates of the model with those of the standard pairing model are always large, especially for the ground and the first excited state. However, the quantum phase crossover in the non-separable pairing model cannot be accounted for by the standard pairing interaction.

Stieltjes polynomials and related quadrature formulae for a class of weight functions, II

Consider a (nonnegative) measure $$d\sigma $$ with support in the interval [a, b] such that the respective orthogonal polynomials satisfy a three-term recurrence relation with coefficients $$\alpha _{n}=\left\{ \begin{array}{ll} \alpha _{e},&{}n~{\hbox {even,}}\\ \alpha _{o},&{}n~ {\hbox {odd,}} \end{array} \right. \beta _{n}=\beta ~{\hbox {for}}~ n\ge \ell $$ , where $$\alpha _{e}, \alpha _{o}, \beta $$ and $$\ell $$ are specific constants. We show that the corresponding Stieltjes polynomials, above the index $$2\ell -1$$ , have a very simple and useful representation in terms of the orthogonal polynomials. As a result of this, the Gauss–Kronrod quadrature formula for $$d\sigma $$ has all the desirable properties, namely, the interlacing of nodes, their inclusion in the closed interval [a, b] (under an additional assumption on $$d\sigma $$ ), and the positivity of all weights, while the formula enjoys an elevated degree of exactness. Furthermore, the interpolatory quadrature formula based on the zeros of the Stieltjes polynomials has positive weights and also elevated degree of exactness. It turns out that this formula is the anti-Gaussian formula for $$d\sigma $$ , while the resulting averaged Gaussian formula coincides with the Gauss–Kronrod formula for this measure. Moreover, we show that the only positive and even measure $$d\sigma $$ on $$(-a,a)$$ for which the Gauss–Kronrod formula is almost of Chebyshev type, i.e., it has almost all of its weights equal, is the measure $$d\sigma (t)=(a^{2}-t^{2})^{-1/2}dt$$ .

An operator approach to the indefinite Stieltjes moment problem

A function f meromorphic on ℂ\ℝ is said to be in the generalized Nevanlinna class N κ (κ ϵ ℤ+), if f is symmetric with respect to ℝ and the kernel $$ {\mathbf{N}}_{\omega }(z)\coloneq \frac{f(z)-\overline{f\left(\omega \right)}}{z-\overline{\omega}} $$ has κ negative squares on ℂ+. The generalized Stieltjes class $$ {\mathbf{N}}_{\kappa}^k\left(\kappa, k\in {\mathrm{\mathbb{Z}}}_{+}\right) $$ is defined as the set of functions f ϵ N κ such that z f ϵ N k . The full indefinite Stieltjes moment problem $$ {MP}_{\kappa}^k\left(\mathbf{s}\right) $$ consists in the following: Given κ, k ϵ ℤ+, and a sequence $$ \mathbf{s}={\left\{{s}_i\right\}}_{i=0}^{\infty } $$ of real numbers, to describe the set of functions $$ f\in {\mathbf{N}}_{\kappa}^k $$ , which satisfy the asymptotic expansion $$ f(z)=-\frac{s_0}{z}-\cdots -\frac{s_2n}{z^{2n+1}}+o\left(\frac{1}{z^{2n+1}}\right)\kern1em \left(z=-y\in {\mathrm{\mathbb{R}}}_{-},y\uparrow \infty \right) $$ for all n big enough. In the present paper, we will solve the indefinite Stieltjes moment problem $$ {MP}_{\kappa}^k\left(\mathbf{s}\right) $$ within the M. G. Krein theory of u-resolvent matrices applied to a Pontryagin space symmetric operator A [0;N] generated by $$ {\mathfrak{J}}_{\left[0;N\right]} $$ . The u-resolvent matrices of the operator A [0;N] are calculated in terms of generalized Stieltjes polynomials, by using the boundary triple’s technique. Some criteria for the problem $$ {MP}_{\kappa}^k\left(\mathbf{s}\right) $$ to be solvable and indeterminate are found. Explicit formulae for Pade approximants for the generalized Stieltjes fraction in terms of generalized Stieltjes polynomials are also presented.

Finite Blaschke products with prescribed critical points, Stieltjes polynomials, and moment problems

The determination of a finite Blaschke product from its critical points is a well-known problem with interrelations to several other topics. Though existence and uniqueness of solutions are established for long, we present new aspects which have not yet been explored to their full extent. In particular, we show that the following three problems are equivalent: (i) determining a finite Blaschke product from its critical points, (ii) finding the equilibrium position of moveable point charges interacting with a special configuration of fixed charges, and (iii) solving a moment problem for the canonical representation of power moments on the real axis. These equivalences are not only of theoretical interest, but also open up new perspectives for the design of algorithms. For instance, the second problem is closely linked to the determination of certain Stieltjes and Van Vleck polynomials for a second order ODE and characterizes solutions as global minimizers of an energy functional.

Anti-Gaussian quadrature formulae based on the zeros of Stieltjes polynomials

It is well known that the Gauss–Kronrod quadrature formula does not always exist with real and distinct nodes and positive weights. In 1996, in an attempt to find an alternative to the Gauss–Kronrod formula for estimating the error of the Gauss quadrature formula, Laurie constructed the anti-Gaussian quadrature formula, which always has real and distinct nodes and positive weights. First, we give a description and prove the most important properties of the anti-Gaussian formula, by applying a different approach than that of Laurie. Then, we consider a measure such that the respective (monic) orthogonal polynomials, above a specific index, satisfy a three-term recurrence relation with constant coefficients. We show that for a measure of this kind the nodes of the anti-Gaussian formula are the zeros of the respective Stieltjes polynomial, while the resulting averaged Gaussian quadrature formula is precisely the corresponding Gauss–Kronrod formula, having elevated degree of exactness. Moreover, we show, by a new method, that a symmetric Gauss–Lobatto quadrature formula is a modified anti-Gaussian formula, and we specialize our results to the measures with constant recurrence coefficients.

Modified Stieltjes polynomials and Gauss–Kronrod quadrature rules

Modified Stieltjes polynomials are defined and used to construct suboptimal extensions of Gaussian rules with one or two degrees less of polynomial exactness than the corresponding Kronrod extension. We prove that, for wide classes of weight functions and a sufficiently large number of nodes, the extended quadratures have positive weights and simple nodes on the interval $$[-1,1]$$ . The classes of weight functions considered complement those for which the Gauss–Kronrod rule is known to exist. Also, strong asymptotic representations on the whole interval $$[-1,1]$$ are given for the modified Stieltjes polynomials, which prove that they behave asymptotically as orthogonal polynomials. Finally, we provide some numerical examples.

A truncated indefinite Stieltjes moment problem

A truncated indefinite Stieltjes moment problem in the class $$ {\mathrm{N}}_{\kappa}^k $$ of generalized Stieltjes functions is studied. The set of solutions of the Stieltjes moment problem is described by the Schur stepby-step algorithm, which is based on the expansion of the solutions in a generalized Stieltjes continued fraction. The resolvent matrix is represented in terms of generalized Stieltjes polynomials. A factorization formula for the resolvent matrix is found.

Exact solution of the two-axis countertwisting hamiltonian for the half-integer J case

Bethe ansatz solutions of the two-axis countertwisting Hamiltonian for any (integer and half-integer) J are derived based on the Jordan–Schwinger (differential) boson realization of the SU(2) algebra after desired Euler rotations, where J is the total angular momentum quantum number of the system. It is shown that solutions to the Bethe ansatz equations can be obtained as zeros of the extended Heine–Stieltjes polynomials. Two sets of solutions, with solution number being J + 1 and J respectively when J is an integer and J + 1/2 each when J is a half-integer, are obtained. Properties of the zeros of the related extended Heine–Stieltjes polynomials for half-integer J cases are discussed. It is clearly shown that double degenerate level energies for half-integer J are symmetric with respect to the E = 0 axis. It is also shown that the excitation energies of the ‘yrast’ and other ‘yrare’ bands can all be asymptotically given by quadratic functions of J, especially when J is large.

Explicit expressions of the generalized Stieltjes polynomial

The existence and uniqueness of a Kronrod type extension to the well-known Gauss-Turan quadrature formulas were proved by Li (1994, pp.71-83). For the generalized Chebyshev weight functions and for the Gori-Micchelli weight function, we found explicit formulas of the corresponding generalized Stieltjes polynomials. General real Kronrod extensions of the Gaussian quadrature formulas with multiple nodes are introduced. In some cases, the explicit expressions of the polynomials, whose zeros are the nodes of the considered quadratures, are determined.

Exact solution of the two-axis countertwisting Hamiltonian

It is shown that the two-axis countertwisting Hamiltonian is exactly solvable when the quantum number of the total angular momentum of the system is an integer after the Jordan–Schwinger (differential) boson realization of the SU(2) algebra. Algebraic Bethe ansatz is used to get the exact solution with the help of the SU(1,1) algebraic structure, from which a set of Bethe ansatz equations of the problem is derived. It is shown that solutions of the Bethe ansatz equations can be obtained as zeros of the Heine–Stieltjes polynomials. The total number of the four sets of the zeros equals exactly 2J+1 for a given integer angular momentum quantum number J, which proves the completeness of the solutions. It is also shown that double degeneracy in level energies may also occur in the J→∞ limit for integer J case except a unique non-degenerate level with zero excitation energy.

An exact solution of spherical mean-field plus a special separable pairing model

An exact solution of nuclear spherical mean-field plus a special orbit-dependent separable pairing model is studied, of which the separable pairing interaction parameters are obtained by a linear fitting in terms of the single-particle energies considered. The advantage of the model is that, similar to the standard pairing case, it can be solved easily by using the extended Heine–Stieltjes polynomial approach. With the analysis of the model in the ds- and pf-shell subspace, it is shown that this special separable pairing model indeed provides similar pair structures of the model with the original separable pairing interaction, and is obviously better than the standard pairing model in many aspects.

Numerical algorithm for the standard pairing problem based on the Heine–Stieltjes correspondence and the polynomial approach

We present a detailed study of the computational complexity of a numerical algorithm based on the Heine–Stieltjes correspondence following the new approach we proposed recently for solving the Bethe ansatz (Gaudin–Richardson) equations of the standard pairing problem. For k pairs of valence nucleons in n non-degenerate single-particle energy levels, the approach utilizes that solutions of the Bethe ansatz equations can be obtained from two matrices of dimensions (k+1)×(k+1) and (n−1)×(k+1), which are associated with the extended Heine–Stieltjes and Van Vleck polynomials, respectively. Since the coefficients in these polynomials are free from divergence with variations in contrast to the original Bethe ansatz equations, the approach provides an efficient and systematic way to solve the problem. The method reduces to solving a system of k polynomial equations, which can be efficiently implemented by the fast Newton–Raphson algorithm with a Monte Carlo sampling procedure for the initial guesses. By extension, the present algorithm can also be used to solve a large class of Gaudin-type quantum many-body problems, including an efficient angular momentum projection method for multi-particle systems. Program summaryProgram title: exactPairingHSCatalogue identifier: AETD_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AETD_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 5411No. of bytes in distributed program, including test data, etc.: 39758Distribution format: tar.gzProgramming language: Mathematica.Computer: Laptop, workstation.Operating system: Tested with MATHEMATICA version 9.0 on Mac OS X and Windows 7.RAM: Less than 10 MBClassification: 17.15.Nature of problem:The program calculates exact pairing energies based on the Heine–Stieltjes polynomial approach. Existing conventional exact-pairing approaches require solving systems of highly nonlinear equations, which are difficult and often impossible to solve beyond the simplest of the quantum-mechanical many-particle systems. In this study, the Heine–Stieltjes polynomial approach is employed to provide solutions for more than one or two pairs of particles residing in many energy levels.Solution method:The new Heine–Stieltjes polynomial approach transforms the pairing problem to one that involves the handling of only two matrix equations. This, combined with an efficient numerical algorithm implemented by the fast Newton–Raphson method with a Monte Carlo sampling procedure for the initial guesses, makes exact pairing solutions feasible even when more energy levels or heavy nuclei (many pairs) are considered.Running time:Less than a hundred seconds using a 2.80 GHz processor. The notebook takes approximately 23 min to complete.

Higher-Order Hermite-Fejér Interpolation for Stieltjes Polynomials

Let and be the ultraspherical polynomials with respect to . Then, we denote the Stieltjes polynomials with respect to satisfying . In this paper, we consider the higher-order Hermite-Fej&#xe9;r interpolation operator based on the zeros of and the higher order extended Hermite-Fej&#xe9;r interpolation operator based on the zeros of . When is even, we show that Lebesgue constants of these interpolation operators are and , respectively; that is, and . In the case of the Hermite-Fej&#xe9;r interpolation polynomials for , we can prove the weighted uniform convergence. In addition, when is odd, we will show that these interpolations diverge for a certain continuous function on , proving that Lebesgue constants of these interpolation operators are similar or greater than log .

The Heine–Stieltjes correspondence and the polynomial approach to the Gaudin–Richardson models

The Heine–Stieltjes correspondence is extended and applied to solve Bethe ansatz equations of the Gaudin–Richardson models, from which the extended Heine–Stieltjes polynomial approach to these models is proposed. As examples for the application of this approach, exact solutions of the standard two-site Bose–Hubbard model and the standard pairing model for nuclei are formulated from the corresponding polynomials.