Nonclassical Polynomials Research Articles

Solvable random-matrix ensemble with a logarithmic weakly confining potential.

This work identifies a solvable (in the sense that spectral correlation functions can be expressed in terms of orthogonal polynomials), rotationally invariant random matrix ensemble with a logarithmic weakly confining potential. The ensemble, which can be interpreted as a transformed Jacobi ensemble, is in the thermodynamic limit characterized by a Lorentzian eigenvalue density. It is shown that spectral correlation functions can be expressed in terms of the nonclassical Gegenbauer polynomials C_{n}^{(-1/2)}(x) with n≥2, which have been proven to form a complete orthogonal set with respect to the proper weight function. A procedure to sample matrices from the ensemble is outlined and used to provide a numerical verification for some of the analytical results. This ensemble is pointed out to potentially have applications in quantum many-body physics.

Non-classical polynomials and the inverse theorem

AbstractIn this paper we characterize when non-classical polynomials are necessary in the inverse theorem for the Gowers $U^k$ -norm. We give a brief deduction of the fact that a bounded function on $\mathbb F_p^n$ with large $U^k$ -norm must correlate with a classical polynomial when $k\le p+1$ . To the best of our knowledge, this result is new for $k=p+1$ (when $p>2$ ). We then prove that non-classical polynomials are necessary in the inverse theorem for the Gowers $U^k$ -norm over $\mathbb F_p^n$ for all $k\ge p+2$ , completely characterising when classical polynomials suffice.

A numerical analysis of motion in symmetric double-well harmonic potentials using pseudospectral methods

I use pseudospectral methods defined with a set of Hermite polynomials and with a set of half-range, nonclassical Hermite polynomials to test their applicability in analyzing the motion of a particle in symmetric double-well harmonic potentials. The results of these two kinds of pseudospectral method and those of the Wentzel-Kramers-Brillouin (WKB) approximation are compared with each other. The pseudospectral method with half-range Hermite polynomials provides more accurate results of the eigenvalues of the five lowest states in symmetric double-well harmonic potentials than that with classical polynomials. It is shown that the former method with a set of the nonclassical polynomials exhibits a faster convergence than the latter and has a feasibility to study motion in a variety of double-well potentials.

Differential Equations for Classical and Non-Classical Polynomial Sets: A Survey

By using the monomiality principle and general results on Sheffer polynomial sets, the differential equation satisfied by several old and new polynomial sets is shown.

Diffusion in a bistable system: The eigenvalue spectrum of the Fokker-Planck operator and Kramers' reaction rate theory.

The time-dependent solution of the Fokker-Planck equation with bistable potentials is considered in terms of the eigenfunctions and eigenvalues of the linear Fokker-Planck operator. The Fokker-Planck equation is the high friction limit of the corresponding Kramers' equation. Two different potentials are considered defined with a constant diffusion coefficient, ε, and position-dependent drift coefficients. The smallest nonzero eigenvalue of the Fokker-Planck operator, λ_{1}, provides the long-time rate coefficient for the transformation of the different species in the two stable states. A novel pseudospectral method with nonclassical polynomials is applied to this class of systems. The convergence of the eigenvalues and eigenfunctions of the Fokker-Planck operator versus the number of basis functions is studied and compared with previous results. The results are consistent with Kramers' theory, and a linear relationship between lnλ_{1} and 1/ε for sufficiently small ε values is verified. A comparison with analytic approximations to λ_{1} is provided.

A Note on Extraction of Orthogonal Polynomials from Generating Function for Reciprocal of Odd Numbers

Motivated by a recent paper by Shashikala [Indian J. Pure Appl. Math. 48 (2) (2017), 177-185] on the extraction four sequences of orthogonal polynomials from generating function from reciprocal of odd numbers, in this note we identify the weight functions and the intervals of orthogonality of these sequences of polynomials. Two of these sequences can be expressed in terms of particular Jacobi polynomials transformed to [0, 1], and other two are non-classical polynomials also orthogonal on [0, 1].

Ultraspherical moments on a set of disjoint intervals

Moment evaluations are important for the study of non-classical orthogonal polynomial systems for which explicit representations are not known. In this paper we compute, in terms of the hypergeometric function, the moments associated with a generalized ultraspherical weight on a collection of intervals with two symmetric gaps. These moments, parametrized by the endpoints of the gaps, are identified as a one parameter deformation between the full range ultraspherical moments and the half range ultraspherical moments.

A Pseudospectral solution of a bistable Fokker–Planck equation that models protein folding

We consider the one-dimensional bistable Fokker–Planck equation proposed by Polotto et al. (2018), with specific drift and diffusion coefficients so as to model protein folding. In this paper, a pseudospectral method is used to solve the Fokker–Planck equation in terms of the eigenvalues (λn) and eigenfunctions (ψn) of the Fokker–Planck operator. Nonclassical polynomials, constructed orthogonal with respect to the equilibrium distribution of the Fokker–Planck equation, are used as basis functions. The eigenvalues determined with the pseudospectral method are compared with the Wentzel–Kramers–Brillouin (WKB) and the SUperSYmmetric (SUSY) Wentzel–Kramers–Brillouin (SWKB) approximations. The eigenvalues calculated differ significantly from those reported by Polotto et al. A detailed study of the role of the lowest non-zero eigenvalue, λ1, to model the rate coefficient for the transition between the bistable states is provided.

Pseudospectral solution of the Schrödinger equation for the Rosen-Morse and Eckart potentials

Pseudospectral methods based on non-classical quadratures are used to numerically compute the eigenvalues and eigenfunctions of the Schrodinger equation for the Rosen-Morse and Eckart potentials. The method uses a basis set of non-classical polynomials, $$\{ P_n(x) \}$$ , orthonormal with respect to a weight function, $$w(x)>0$$ , to construct an $$N \times N$$ matrix representative, $$\{ H_{nm} \}$$ , of the Hamiltonian, H. This matrix representative is transformed to an equivalent pseudospectral representative, $$\{H_{ij}\}$$ . The rate of convergence of the eigenvalues of $$\{ H_{ij} \}$$ to the eigenvalues of H, versus the grid size N, is reported for non-classical basis functions in comparison with the use of Legendre and Laguerre polynomials as well as a Fourier basis. The use of non-classical polynomials is shown to provide the fastest convergence for the eigenvalues. The pseudospectral method based on nonclassical quadratures proposed in this paper should find wide applicability to other problems in quantum and statistical mechanics. A review is provided of the use of a tensor product of one dimensional basis functions to describe two and three dimensional problems.

Pseudospectral method of solution of the Schrödinger equation for the Kratzer and pseudoharmonic potentials with nonclassical polynomials and applications to realistic diatom potentials

The pseudoharmonic and Kratzer potentials have been extensively employed by numerous workers to model the vibrational states of diatomic molecules. These potentials belong to SUperSYmmetric (SUSY) quantum mechanics and the eigenvalues of the Schrödinger equation are known. The energy eigenvalues of the Schrödinger equation for the pseudoharmonic and Kratzer potentials have been determined with different numerical methods as a benchmark of the numerical schemes. We employ a pseudospectral method based on a quadrature grid defined with a nonclassical polynomial basis set. This basis set is defined orthogonal with respect to the square of ground state wavefunction as the weight function. A discrete matrix representation of the Hamiltonian of dimension N is constructed and vibrational energies are calculated with the numerical diagonalization of this matrix. This pseudospectral method is employed to calculate the vibrational energy levels of H2, CO and NO modelled with the pseudoharmonic and Kratzer potentials in comparison with the more realistic Morse potential. The vibrational energy eigenstates for H2 with a realistic quantum mechanical potential and an approximate Morse potential are also calculated. The oversimplified pseudoharmonic and Kratzer potentials are useful for benchmarking numerical methods. However, they represent potential energy curves that can deviate drastically from more exact potentials such as the Morse potential.

Center of mass distribution of the Jacobi unitary ensembles: Painlevé V, asymptotic expansions

In this paper, we study the probability density function, P(c,α,β,n) dc, of the center of mass of the finite n Jacobi unitary ensembles with parameters α > −1 and β > −1; that is the probability that trMn ∈ (c, c + dc), where Mn are n × n matrices drawn from the unitary Jacobi ensembles. We compute the exponential moment generating function of the linear statistics ∑j=1n f(xj)≔∑j=1nxj, denoted by Mf(λ,α,β,n). The weight function associated with the Jacobi unitary ensembles reads xα(1 − x)β, x ∈ [0, 1]. The moment generating function is the n × n Hankel determinant Dn(λ, α, β) generated by the time-evolved Jacobi weight, namely, w(x; λ, α, β) = xα(1 − x)β e−λx, x ∈ [0, 1], α > −1, β > −1. We think of λ as the time variable in the resulting Toda equations. The non-classical polynomials defined by the monomial expansion, Pn(x, λ) = xn + p(n, λ) xn−1 + ⋯ + Pn(0, λ), orthogonal with respect to w(x, λ, α, β) over [0, 1] play an important role. Taking the time evolution problem studied in Basor et al. [J. Phys. A: Math. Theor. 43, 015204 (2010)], with some change of variables, we obtain a certain auxiliary variable rn(λ), defined by integral over [0, 1] of the product of the unconventional orthogonal polynomials of degree n and n − 1 and w(x; λ, α, β)/x. It is shown that rn(2iez) satisfies a Chazy II equation. There is another auxiliary variable, denoted as Rn(λ), defined by an integral over [0, 1] of the product of two polynomials of degree n multiplied by w(x; λ, α, β)/x. Then Yn(−λ) = 1 − λ/Rn(λ) satisfies a particular Painlevé V: PV(α2/2, − β2/2, 2n + α + β + 1, 1/2). The σn function defined in terms of the λp(n, −λ) plus a translation in λ is the Jimbo–Miwa–Okamoto σ-form of Painlevé V. The continuum approximation, treating the collection of eigenvalues as a charged fluid as in the Dyson Coulomb Fluid, gives an approximation for the moment generating function Mf(λ,α,β,n) when n is sufficiently large. Furthermore, we deduce a new expression of Mf(λ,α,β,n) when n is finite, in terms the σ function of this is a particular case of Painlevé V. An estimate shows that the moment generating function is a function of exponential type and of order n. From the Paley-Wiener theorem, one deduces that P(c,α,β,n) has compact support [0, n]. This result is easily extended to the β ensembles, as long as the weight w is positive and continuous over [0, 1].

A comparison of pseudospectral methods for the solution of the Schrödinger equation; the Lennard-Jones (n, 6) potential

The eigenvalues of the Schrödinger equation for the Lennard-Jones (n,6) potential with n=8, 10 and 12, have been determined by numerous workers with different numerical methods. In this paper, we consider a pseudospectral method based on a grid defined by the quadrature points associated with nonclassical polynomials. The polynomials are orthogonal with respect to the weight function defined by the ground state wave function for a Morse potential that approximates the Lennard-Jones potential. The Morse potential, unlike the Lennard-Jones potential, belongs to the class of potentials in SUperSYmmetric (SUSY) quantum mechanics. The ground state wavefunction is thus known and used as the weight function to define the nonclassical grid. An alternate supersymmeric weight function that approximates the Lennard-Jones potential is also considered. The results obtained in this way are compared with the results obtained with a direct numerical integration of the radial Schrödinger equation, a pseudospectral Fourier method and Jeffreys-Wentzel-Kramers-Brouillon (JWKB) approximations as well as other approximate analytical methods. The convergence of the eigenvalues with the nonclassical quadrature grid is demonstrated to be rapid and very competitive with the other methods.

Lower Bounds for Constant Query Affine-Invariant LCCs and LTCs

Affine-invariant codes are codes whose coordinates form a vector space over a finite field and which are invariant under affine transformations of the coordinate space. They form a natural, well-studied class of codes; they include popular codes such as Reed-Muller and Reed-Solomon. A particularly appealing feature of affine-invariant codes is that they seem well suited to admit local correctors and testers. In this work, we give lower bounds on the length of locally correctable and locally testable affine-invariant codes with constant query complexity. We show that if a code C ⊂ Σ K n is an r -query affine invariant locally correctable code (LCC), where K is a finite field and Σ is a finite alphabet, then the number of codewords in C is at most exp( O K,r,|Σ| ( n r −1)) . Also, we show that if C ⊂ Σ K n is an r -query affine invariant locally testable code (LTC), then the number of codewords in C is at most exp( O K,r,|Σ| ( n r −2)) . The dependence on n in these bounds is tight for constant-query LCCs/LTCs, since Guo, Kopparty, and Sudan (ITCS’13) constructed affine-invariant codes via lifting that have the same asymptotic tradeoffs. Note that our result holds for non-linear codes, whereas previously, Ben-Sasson and Sudan (RANDOM’11) assumed linearity to derive similar results. Our analysis uses higher-order Fourier analysis. In particular, we show that the codewords corresponding to an affine-invariant LCC/LTC must be far from each other with respect to Gowers norm of an appropriate order. This then allows us to bound the number of codewords, using known decomposition theorems, which approximate any bounded function in terms of a finite number of low-degree non-classical polynomials, up to a small error in the Gowers norm.

New Properties of the Zeros of Krall Polynomials

We identify a new class of algebraic relations satisfied by the zeros of orthogonal polynomials that are eigenfunctions of linear differential operators of order higher than two, known as Krall polynomials. Given an orthogonal polynomial family \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{ pv(x)\\} \\frac{\\infty }{v}0$$\\end{document}, we relate the zeros of the polynomial pN with the zeros of pm for each m ≤ N (the case m = N corresponding to the relations that involve the zeros of pN only). These identities are obtained by finding exact expressions for the similarity transformation that relates the spectral and the (interpolatory) pseudospectral matrix representations of linear differential operators, while using the zeros of the polynomial pN as the interpolation nodes. The proposed framework generalizes known properties of classical orthogonal polynomials to the case of nonclassical polynomial families of Krall type. We illustrate the general result by proving new identities satisfied by the Krall-Legendre, the Krall-Laguerre and the Krall-Jacobi orthogonal polynomials.

Pseudospectral Solution of the Fokker–Planck Equation with Equilibrium Bistable States: the Eigenvalue Spectrum and the Approach to Equilibrium

The Fokker–Planck equation with a constant diffusion coefficient and a particular polynomial drift coefficient can exhibit a bistable equilibrium distribution. Such model systems have been used to study chemical reactions, nucleation, climate, optical bistability and other phenomena. In this paper, we consider a particular choice for the drift coefficient of the form $$A(x) = x^5 - x^3$$ to exemplify the statistical behaviour of such systems. The transformation of the Fokker–Planck equation to a Schrodinger equation leads to a potential that belongs to the class of potentials in supersymmetric (SUSY) quantum mechanics. A pseudospectral method based on nonclassical polynomials is used to determine the spectrum of the Fokker–Planck operator and of the Schrodinger equation. The converged numerical eigenvalues are compared with WKB and SWKB approximations of the eigenvalues.

Pseudospectral method of solution of the Schrödinger equation with non classical polynomials; the Morse and Pöschl–Teller (SUSY) potentials

The eigenvalue spectra of the Schrödinger equation for the Morse and Pöschl–Teller potentials are calculated with three pseudospectral methods. The first method used is based on quadratures defined in terms of nonclassical polynomials orthogonal with a weight function equal to the square of the ground state eigenfunction. This is possible since the Morse and Pöschl–Teller potentials belong to the class of potentials in SUperSYmetric (SUSY) quantum mechanics. The other methods used are a Fourier grid method, often referred to as the discrete variable representation, and the Lagrange mesh method based on the Hermite quadrature. The convergence of the eigenvalues towards the known exact values versus the number of grid points is reported with the three methods for the Morse and Pöschl–Teller potentials. The difficulty to obtain converged results for the highest bound state is discussed.

General systems of linear forms: Equidistribution and true complexity

Higher-order Fourier analysis is a powerful tool that can be used to analyze the densities of linear systems (such as arithmetic progressions) in subsets of Abelian groups. We are interested in the group Fpn, for fixed p and large n, where it is known that analyzing these averages reduces to understanding the joint distribution of a family of sufficiently pseudorandom (formally, high-rank) nonclassical polynomials applied to the corresponding system of linear forms.In this work, we give a complete characterization for these distributions for arbitrary systems of linear forms. This extends previous works which accomplished this in some special cases. As an application, we resolve a conjecture of Gowers and Wolf on the true complexity of linear systems. Our proof deviates from that of the previously known special cases and requires several new ingredients. One of which, which may be of independent interest, is a new theory of homogeneous nonclassical polynomials.

Variational Estimates of the Spectral Gap and Eigenvalue Spectra of the Linearized and Linear Boltzmann Hard Sphere Collision Operators

The eigenvalue spectra of the linearized Boltzmann collision operator and the mass ratio–dependent linear Boltzmann collision operator for binary systems are reported. A variational method is used with either the Sonine polynomial basis set or the basis set of nonclassical Maxwell polynomials orthogonal with respect to the weight function , x ∈ [0, ∞). In the latter case, a pseudospectral method is used based on the Maxwell quadrature weights and points. A multidomain pseudospectral method is also used to accurately calculate the eigenvalues close to the continuum boundary so as to compare with the asymptotic Wentzel-Kramers-Brillouin results.

Accurate spectral numerical schemes for kinetic equations with energy diffusion

We examine the merits of using a family of polynomials that are orthogonal with respect to a non-classical weight function to discretize the speed variable in continuum kinetic calculations. We consider a model one-dimensional partial differential equation describing energy diffusion in velocity space due to Fokker–Planck collisions. This relatively simple case allows us to compare the results of the projected dynamics with an expensive but highly accurate spectral transform approach. It also allows us to integrate in time exactly, and to focus entirely on the effectiveness of the discretization of the speed variable. We show that for a fixed number of modes or grid points, the non-classical polynomials can be many orders of magnitude more accurate than classical Hermite polynomials or finite-difference solvers for kinetic equations in plasma physics. We provide a detailed analysis of the difference in behavior and accuracy of the two families of polynomials. For the non-classical polynomials, if the initial condition is not smooth at the origin when interpreted as a three-dimensional radial function, the exact solution leaves the polynomial subspace for a time, but returns (up to roundoff accuracy) to the same point evolved to by the projected dynamics in that time. By contrast, using classical polynomials, the exact solution differs significantly from the projected dynamics solution when it returns to the subspace. We also explore the connection between eigenfunctions of the projected evolution operator and (non-normalizable) eigenfunctions of the full evolution operator, as well as the effect of truncating the computational domain.

A hidden analytic structure of the Rabi model

The Rabi model describes the simplest interaction between a cavity mode with a frequency ωc and a two-level system with a resonance frequency ω0. It is shown here that the spectrum of the Rabi model coincides with the support of the discrete Stieltjes integral measure in the orthogonality relations of recently introduced orthogonal polynomials. The exactly solvable limit of the Rabi model corresponding to Δ=ω0/(2ωc)=0, which describes a displaced harmonic oscillator, is characterized by the discrete Charlier polynomials in normalized energy ϵ, which are orthogonal on an equidistant lattice. A non-zero value of Δ leads to non-classical discrete orthogonal polynomials ϕk(ϵ) and induces a deformation of the underlying equidistant lattice. The results provide a basis for a novel analytic method of solving the Rabi model. The number of ca. 1350 calculable energy levels per parity subspace obtained in double precision (cca 16 digits) by an elementary stepping algorithm is up to two orders of magnitude higher than is possible to obtain by Braak’s solution. Any first n eigenvalues of the Rabi model arranged in increasing order can be determined as zeros of ϕN(ϵ) of at least the degree N=n+nt. The value of nt>0, which is slowly increasing with n, depends on the required precision. For instance, nt≃26 for n=1000 and dimensionless interaction constant κ=0.2, if double precision is required. Given that the sequence of the lth zeros xnl’s of ϕn(ϵ)’s defines a monotonically decreasing discrete flow with increasing n, the Rabi model is indistinguishable from an algebraically solvable model in any finite precision. Although we can rigorously prove our results only for dimensionless interaction constant κ<1, numerics and exactly solvable example suggest that the main conclusions remain to be valid also for κ≥1.