Terms Of Orthogonal Polynomials Research Articles

Four-Parameter 1/r2 Singular Short-Range Potential with Rich Bound States and A Resonance Spectrum

We use the tridiagonal representation approach to enlarge the class of exactly solvable quantum systems. For this, we use a square-integrable basis in which the matrix representation of the wave operator is tridiagonal. In this case, the wave equation becomes a three-term recurrence relation for the expansion coefficients of the wave function with a solution in terms of orthogonal polynomials that is equivalent to a solution of the original problem. We obtain S-wave bound states for a new four-parameter potential with a 1/r2 singularity but short-range, which has an elaborate configuration structure and rich spectral properties. A particle scattered by this potential must overcome a barrier and can then be trapped in the potential valley in a resonance or bound state. Using complex rotation, we demonstrate the rich spectral properties of the potential in the case of a nonzero angular momentum and show how this structure varies with the parameters of the potential.

Four-parameter $1/r^2$ singular short-range potential with rich bound states and a resonance spectrum

Метод трехдиагонального представления применяется с целью расширить класс точно решаемых квантовых систем, для чего используется квадратично интегрируемый базис, в котором матрица волнового оператора трехдиагональна. При этом волновое уравнение принимает вид рекуррентного соотношения для коэффициентов разложения волновой функции трех последовательных порядков, решение которого в терминах ортогональных полиномов эквивалентно решению исходной задачи. Получены S-волновые связанные состояния для нового четырехпараметрического потенциала с особенностью типа $1/r^2$, но короткодействующего, который обладает сложной конфигурационной структурой и богатыми спектральными свойствами. Частица при рассеянии на таком потенциале должна преодолевать потенциальный барьер, а затем может быть захвачена в ловушку в потенциальной яме в резонансном или связанном состоянии. С использованием комплексного вращения продемонстрированы богатые спектральные свойства потенциала в случае ненулевого углового момента и показано, как эта структура изменяется с изменением параметров потенциала.

Solution of Spin and Pseudo-Spin Symmetric Dirac Equation in (1+1) Space-Time Using Tridiagonal Representation Approach

The aim of this work is to find exact solutions of the Dirac equation in (1+1) space-time beyond the already known class. We consider exact spin (and pseudo-spin) symmetric Dirac equations where the scalar potential is equal to plus (and minus) the vector potential. We also include pseudo-scalar potentials in the interaction. The spinor wavefunction is written as a bounded sum in a complete set of square integrable basis, which is chosen such that the matrix representation of the Dirac wave operator is tridiagonal and symmetric. This makes the matrix wave equation a symmetric three-term recursion relation for the expansion coefficients of the wavefunction. We solve the recursion relation exactly in terms of orthogonal polynomials and obtain the state functions and corresponding relativistic energy spectrum and phase shift.

Matrix orthogonal polynomials associated with perturbations of block Toeplitz matrices

Let {c j } =0 be a sequence of matrix moments associated with a matrix of measures supported on the unit circle, and let {P j } =0 be its corresponding sequence of monic matrix orthogonal polynomials. In this contribution, we consider a perturbation on the moments and find an explicit relation for the perturbed orthogonal polynomials in terms of {P j } =0 . We also obtain an expression for the corresponding second kind polynomials.

Efficient algorithms for the dynamics of large and infinite classical central spin models

We investigate the time dependence of correlation functions in the central spin model, which describes the electron or hole spin confined in a quantum dot, interacting with a bath of nuclear spins forming the Overhauser field. For large baths, a classical description of the model yields quantitatively correct results. We develop and apply various algorithms in order to capture the long-time limit of the central spin for bath sizes from 1000 to infinitely many bath spins. Representing the Overhauser field in terms of orthogonal polynomials, we show that a carefully reduced set of differential equations is sufficient to compute the spin correlations of the full problem up to very long times, for instance up to $10^5\hbar/J_\mathrm{Q}$ where $J_\mathrm{Q}$ is the natural energy unit of the system. This technical progress renders an analysis of the model with experimentally relevant parameters possible. We benchmark the results of the algorithms with exact data for a small number of bath spins and we predict how the long-time correlations behave for different effective numbers of bath spins.

Solution of the nonrelativistic wave equation using the tridiagonal representation approach

We choose a complete set of square integrable functions as a basis for the expansion of the wavefunction in configuration space such that the matrix representation of the nonrelativistic time-independent linear wave operator is tridiagonal and symmetric. Consequently, the matrix wave equation becomes a symmetric three-term recursion relation for the expansion coefficients of the wavefunction. The recursion relation is then solved exactly in terms of orthogonal polynomials in the energy. Some of these polynomials are not found in the mathematics literature. The asymptotics of these polynomials give the phase shift for the continuous energy scattering states and the spectrum for the discrete energy bound states. Depending on the space and boundary conditions, the basis functions are written in terms of either the Laguerre or Jacobi polynomials. The tridiagonal requirement limits the number of potential functions that yield exact solutions of the wave equation. Nonetheless, the class of exactly solvable problems in this approach is larger than the conventional class (see, for example, Table XII in the text). We also give very accurate results for cases where the wave operator matrix is not tridiagonal but its elements could be evaluated either exactly or numerically with high precision.

Inrush Blocking Scheme in Transformer Differential Protection

Transformer inrush currents during energization result in mal operation of transformer differential protection, due to the flow of magnetizing current only on the primary / source side of the transformer. A straightforward approach in recognition of inrush currents is proposed here by checking the core cause of this inrush phenomenon i.e. the magnetic saturation of the transformer core. In this method, the pseudocharacteristics were approximated in terms of orthogonal polynomials, and is used to distinguish between inrush currents and fault currents. In this method, the magnetic saturation characteristics of the transformer is modeled by defining pseudocharacteristics, and is obtained over a half cycle data window of the differential current. This data is assessed using certain criteria to detect inrush and fault currents. This method is not reliant on the transformer models and is precise in its output. The propositioned method was tested using delta-wye transformers under both fault and inrush scenarios.

The Kharitonov theorem and robust stabilization via orthogonal polynomials

Kharitonov’s theorem for interval polynomials is given in terms of orthogonal polynomials on $[0, +\infty )$ and their second kind polynomials. A family of robust stabilizing controls for the canonical system is proposed.

Matrix-valued orthogonal polynomials related to the quantum analogue of (mathrm{SU}(2) times mathrm{SU}(2), mathrm{diag})

Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of (mathrm{SU}(2) times mathrm{SU}(2), mathrm{diag}) are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised universal enveloping algebra with a coideal subalgebra. The matrix-valued spherical functions give rise to matrix-valued orthogonal polynomials, which are matrix-valued analogues of a subfamily of Askey–Wilson polynomials. For these matrix-valued orthogonal polynomials, a number of properties are derived using this quantum group interpretation: the orthogonality relations from the Schur orthogonality relations, the three-term recurrence relation and the structure of the weight matrix in terms of Chebyshev polynomials from tensor product decompositions, and the matrix-valued Askey–Wilson type q-difference operators from the action of the Casimir elements. A more analytic study of the weight gives an explicit LDU-decomposition in terms of continuous q-ultraspherical polynomials. The LDU-decomposition gives the possibility to find explicit expressions of the matrix entries of the matrix-valued orthogonal polynomials in terms of continuous q-ultraspherical polynomials and q-Racah polynomials.

Spectral likelihood expansions for Bayesian inference

A spectral approach to Bayesian inference is presented. It pursues the emulation of the posterior probability density. The starting point is a series expansion of the likelihood function in terms of orthogonal polynomials. From this spectral likelihood expansion all statistical quantities of interest can be calculated semi-analytically. The posterior is formally represented as the product of a reference density and a linear combination of polynomial basis functions. Both the model evidence and the posterior moments are related to the expansion coefficients. This formulation avoids Markov chain Monte Carlo simulation and allows one to make use of linear least squares instead. The pros and cons of spectral Bayesian inference are discussed and demonstrated on the basis of simple applications from classical statistics and inverse modeling.

Transformer Differential Protection by Online Core Modeling and Orthogonal Polynomials

By expressing the newly defined “magnetic pseudo-characteristic” in terms of orthogonal polynomials, a new method is proposed to discriminate between inrush and fault currents. In this method, the operating segment of the magnetic pseudocharacteristic is extracted over a half-cycle data window of differential current, and is then evaluated by some criteria to detect fault/inrush currents. Independence from transformer models and high accuracy are some merits of this method. The method is tested on the simulated data of wye–delta and wye–wye-connected transformers (three limb) and the results show the average fault detection delay of 16 ms (in 50-Hz base) with an accuracy above 97%. Also, for a more practical assessment, the results are compared with those of the well-known second harmonic restraint method, and this verifies greater accuracy of the proposed method. Moreover, performance of the method in noisy environments is studied. Details about efficient implementation by using fast Fourier transform algorithms are also included.

A matrix characterization for the [formula omitted]-semiclassical and [formula omitted]-coherent orthogonal polynomials

We present a new structure relation for the sequence of orthogonal polynomials associated with a Dν-semiclassical linear functional of class s, and then we use it to obtain a matrix characterization of the Dν-semiclassical orthogonal polynomials in terms of the Jacobi matrix associated with the multiplication operator in the basis of orthonormal polynomials, and the nonsingular lower triangular matrix that represents the orthogonal polynomials with respect to some bases of polynomials. We also provide a matrix characterization of Dν-coherent pairs of linear functionals.

The supercritical regime in the normal matrix model with cubic potential

The normal matrix model with a cubic potential is ill-defined and it develops a critical behavior in finite time. We follow the approach of Bleher and Kuijlaars to reformulate the model in terms of orthogonal polynomials with respect to a Hermitian form. This reformulation was shown to capture the essential features of the normal matrix model in the subcritical regime, namely that the zeros of the polynomials tend to a number of segments (the motherbody) inside a domain (the droplet) that attracts the eigenvalues in the normal matrix model.In the present paper we analyze the supercritical regime and we find that the large n behavior is described by the evolution of a spectral curve satisfying the Boutroux condition. The Boutroux condition determines a system of contours Σ1, consisting of the motherbody and whiskers sticking out of the domain. We find a second critical behavior at which the original motherbody shrinks to a point at the origin and only the whiskers remain.In the regime before the second criticality we also give strong asymptotics of the orthogonal polynomials by means of a steepest descent analysis of a 3×3 matrix valued Riemann–Hilbert problem. It follows that the zeros of the orthogonal polynomials tend to Σ1, with the exception of at most three spurious zeros.

The Spectral Connection Matrix for Any Change of Basis within the Classical Real Orthogonal Polynomials

The connection problem for orthogonal polynomials is, given a polynomial expressed in the basis of one set of orthogonal polynomials, computing the coefficients with respect to a different set of orthogonal polynomials. Expansions in terms of orthogonal polynomials are very common in many applications. While the connection problem may be solved by directly computing the change–of–basis matrix, this approach is computationally expensive. A recent approach to solving the connection problem involves the use of the spectral connection matrix, which is a matrix whose eigenvector matrix is the desired change–of–basis matrix. In Bella and Reis (2014), it is shown that for the connection problem between any two different classical real orthogonal polynomials of the Hermite, Laguerre, and Gegenbauer families, the related spectral connection matrix has quasiseparable structure. This result is limited to the case where both the source and target families are one of the Hermite, Laguerre, or Gegenbauer families, which are each defined by at most a single parameter. In particular, this excludes the large and common class of Jacobi polynomials, defined by two parameters, both as a source and as a target family. In this paper, we continue the study of the spectral connection matrix for connections between real orthogonal polynomial families. In particular, for the connection problem between any two families of the Hermite, Laguerre, or Jacobi type (including Chebyshev, Legendre, and Gegenbauer), we prove that the spectral connection matrix has quasiseparable structure. In addition, our results also show the quasiseparable structure of the spectral connection matrix from the Bessel polynomials, which are orthogonal on the unit circle, to any of the Hermite, Laguerre, and Jacobi types. Additionally, the generators of the spectral connection matrix are provided explicitly for each of these cases, allowing a fast algorithm to be implemented following that in Bella and Reis (2014).

A matrix approach for the semiclassical and coherent orthogonal polynomials

We obtain a matrix characterization of semiclassical orthogonal polynomials in terms of the Jacobi matrix associated with the multiplication operator in the basis of orthogonal polynomials, and the lower triangular matrix that represents the orthogonal polynomials in terms of the monomial basis of polynomials. We also provide a matrix characterization for coherent pairs of linear functionals.

Exact Solution of the Curved Dirac Equation in Polar Coordinates: Master Function Approach

We show that the (2+1) curved Dirac equation in polar coordinates can be transformed into Schrodinger-like differential equation for upper spinor component. We compare this equation with the Schrodinger equation derived from shape invariance property of second order differential equations of mathematical physics. This formalism enables us to determine the electrostatic potential and relativistic energy in terms of master function and corresponding weight function. We also obtain the spinor wave function in terms of orthogonal polynomials.

Sampling unitary ensembles

We develop a computationally efficient algorithm for sampling from a broad class of unitary random matrix ensembles that includes but goes well beyond the straightforward to sample Gaussian unitary ensemble (GUE). The algorithm exploits the fact that the eigenvalues of unitary ensembles (UEs) can be represented as a determinantal point process whose kernel is given in terms of orthogonal polynomials. Consequently, our algorithm can be used to sample from UEs for which the associated orthogonal polynomials can be numerically computed efficiently. By facilitating high accuracy sampling of non-classical UEs, the algorithm can aid in the experimentation-based formulation or refutation of universality conjectures involving eigenvalue statistics that might presently be unamenable to theoretical analysis. Examples of such experiments are included.

Orthogonal polynomial approach to estimation of poles of rational functions from data on open curves

This paper deals with the problem of finding poles of rational functions from function values on open curves in the complex plane. For this problem, Nara and Ando recently proposed an algorithm that reduces the problem to a system of linear equations through contour integration. The main aim of this paper is to analyze and improve this algorithm by giving a new interpretation to the algorithm in terms of orthogonal polynomials. It is demonstrated that the system of linear equations is not always uniquely solvable and that this difficulty can be remedied by doubling the number of the linear equations. Moreover, to cope with discretization errors caused by numerical integration, we introduce new polynomials similar, in spirit, to discrete orthogonal polynomials, which yield an algorithm free from discretization errors.

Propagation of nonclassical states of light through one-dimensional photonic lattices

We study the propagation of non-classical light through arrays of coupled linear photonic waveguides and introduce some sets of refractive indices and coupling parameters that provide a closed form propagator in terms of orthogonal polynomials. We present propagation examples of non-classical states of light: single photon, coherent state, path-entangled state and two-mode squeezed vacuum impinging into two-waveguide couplers and a photonic lattice producing coherent transport.

Dirac’s formalism combined with complex Fourier operational matrices to solve initial and boundary value problems

Approximations of functions in terms of orthogonal polynomials have been used to develop and implement numerical approaches to solve spectrally initial and boundary value problems. The main idea behind these approaches is to express differential and integral operators by using matrices, and this, in turn, makes the numerical implementation easier to be expressed in computational algebraic languages. In this paper, the application of the methodology is enlarged by using Dirac’s formalism, combined with complex Fourier series.