Zeros Of Laguerre Polynomials Research Articles

Generalized adiabatic approximation to the quantum Rabi model

The quantum Rabi model (QRM) describes the interaction between a two-level system (qubit) and a quantum harmonic oscillator. In the limit where the qubit frequency is smaller than the harmonic frequency, the QRM can be well approximated by the adiabatic approximation (AA). The AA is widely used due to its simplicity and explicit physical interpretation. However, the level crossings in the spectrum of the QRM predicted by the AA are determined by the zeros of Laguerre polynomials, which deviate from the exact points. We propose a new approximation to the QRM that predicts the level crossings correctly. This is done by exploiting a surprising connection between isolated exact solutions to the QRM and the Laguerre polynomials in the AA. We thus refer to this approach as the generalized adiabatic approximation (GAA). By construction, the GAA always predicts the exact exceptional spectrum and approximates the regular spectrum remarkably well in a much larger parameter regime than the AA. This generalized approach offers a framework to deal with the family of Rabi-type light-matter interaction models in a simple but accurate manner.

Weighted Pál type (1;0)-interpolation on the zeros of Laguerre abscissas

In this paper, we have studied a Pál type (1; 0)-interpolation when first derivatives and function values are prescribed on the zeros of Laguerre Polynomials $L_{n}^{(\alpha)}(x), \alpha > -1$ and its derivative $(L_{n}^{(\alpha)})^{\prime}(x)$ respectively. Existence, uniqueness, explicit representation and a quantitative estimate of the interpolatory polynomial $R_{n, \alpha}(x)$ has been obtained.

Subordination by conformal martingales in Lp and zeros of Laguerre polynomials

Given martingales W and Z such that W is differentially subordinate to Z, Burkholder obtained the sharp inequality E|W|p≤(p∗−1)pE|Z|p, where p∗=max {p,p/(p−1)}. What happens if one of the martingales is also a conformal martingale? Bañuelos and Janakiraman proved that if p≥2 and W is a conformal martingale differentially subordinate to any martingale Z, then E|W|p≤[(p2−p)/2]p/2E|Z|p. In this paper, we establish that if p≥2, Z is conformal, and W is any martingale subordinate to Z, then E|W|p≤[2(1−zp)/zp]pE|Z|p, where zp is the smallest positive zero of a certain solution of the Laguerre ordinary differential equation. We also prove the sharpness of this estimate and an analogous one in the dual case for 1<p<2. Finally, we give an application of our results. Previous estimates on the Lp-norm of the Beurling–Ahlfors transform give at best ‖B‖p≲2p as p→∞. We improve this to ‖B‖p≲1.3922p as p→∞.

Ground state of 1D bosons with delta interaction: link to the BCS model

The Bethe roots describing the ground state energy of the integrable 1D model of interacting bosons with weakly repulsive two-body delta interactions are seen to satisfy the set of Richardson equations appearing in the strong coupling limit of an integrable BCS pairing model. The BCS model describes boson-boson interactions with zero centre of mass momentum of pairs. It follows that the Bethe roots of the weakly interacting boson model are given by the zeros of Laguerre polynomials. The ground state energy and the lowest excitation are obtained explicitly via the Bethe roots. A direct link has thus been established, in the context of integrable 1D models, between bosons interacting via weakly repulsive two-body delta-interactions and strongly interacting Cooper pairs of bosons.

The unexplained accuracy of the Lagrange-mesh method.

The Lagrange-mesh method is an approximate variational calculation which resembles a mesh calculation because of the use of a Gauss quadrature. In order to analyze its accuracy, four different Lagrange-mesh calculations based on the zeros of Laguerre polynomials are compared with exact variational calculations based on the corresponding Laguerre basis. The comparison is performed for three solvable radial potentials: the Morse, harmonic-oscillator, and Coulomb potentials. The results show that the accuracies of the energies obtained for different partial waves with the different mesh approximations are very close to the variational accuracy, even in the presence of the centrifugal singularity. The same property holds for the approximate wave functions. This striking accuracy remains unexplained.

Approximation of the Hilbert Transform on the real semiaxis using Laguerre zeros

The authors propose two new algorithms for the computation of Cauchy principal value integrals on the real semiaxis. The proposed quadrature rules use zeros of Laguerre polynomials. Theoretical error estimates are proved and some numerical examples are showed.

Gaussian quadrature as a numerical integration method for estimating area under the curve.

This paper presents a numerical integration method for estimating the area under the curve (AUC) over the infinite time interval. This method is based on the Gauss-Laguerre quadrature and produces AUC estimates over the infinite time interval without extrapolation in a usual sense. By contrast, in traditional schemes, piecewise interpolation is used to obtain the area up to the final sampling point, and the remaining portion is extrapolated using nonlinear regression. In this case, there is no theoretical consistency between the quadrature and extrapolation. The inconsistency may cause certain problems. For example, the optimal sampling criterion for the former is not necessarily optimal for the latter. Such inconsistency does not arise in the method of this work. The sampling points are placed near the zeros of Laguerre polynomials so as to directly estimate the AUC over the infinite time interval. The sampling design requires no particular prior information. This is also advantageous over the previous strategy, which worked by minimizing the variance of estimated AUC under the assumptions of particular pharmacokinetic and variance functions. The original Gaussian quadrature is believed to be inappropriate for numerical integration of data because of several restrictions. In this paper, it is shown that, using a simple strategy for managing errors due to these restrictions, the method produces an estimate of AUC with practically sufficient precision. The efficacy of this method is finally shown by numerical simulations in which the bias and variance of its estimate were compared with those of the previous methods such as the trapezoidal, log-trapezoidal, Lagrange, and parabolas-through-the-origin methods.

Lagrange-mesh calculations of three-body atoms and molecules

The Lagrange-mesh numerical method has the simplicity of a mesh calculation and the accuracy of a variational calculation. A three-dimensional Lagrange-mesh method based on zeros of Laguerre polynomials is applied to the study of the ground states of the helium atom, the hydrogen and positronium negative ions and the hydrogen molecular ion. The calculations lead to big sparse matrices. Highly accurate energies are obtained, which essentially agree with recently published results or improve them. The accuracy is estimated by varying the mesh size and two nonlinear scale parameters. Different mean values of observables are obtained very simply with the Gauss quadrature approximation associated with the mesh.

Nystroem interpolants based on zeros of Laguerre polynomials for some Weiner-Hopf equations

We consider a special class of Wiener-Hopf integral equations u(y) - λ ∫ 0 ∞ h(y - x)u(x) dx = f(y), with || ∫ 0 ∞ |h(y - x)|dx||∞ < λ -1 , whose solutions u(x) decay exponentially at infinity. First we rewrite the equation in the form u(y) - μe Y ∫ 0 ∞ e -x k(y - x)v(x) dx = g(y), with ||e Y ∫ 0 ∞ e -x |k(y - x)|dx||∞ < μ -1 . Then we propose to solve this equation by using a Nystrom interpolant based on a product rule of the type ∫ 0 ∞ e -x k(y - x)v(x) dx formula math ω mi (y)v(x mi ), where the nodes {x mi } are the zeros of the m-degree Laguerre polynomial. For these interpolants we prove stability in C[0, ∞) and derive uniform convergence estimates.

Convergence of product integration rules over (0,∞) for functions with weak singularities at the origin

In this paper we consider integrals of the form\[∫0∞e−xK(x,y)f(x)dx,\int _0^\infty {{e^{ - x}}K(x,y)f(x)dx,}\]withf∈Cp[0,∞)∩Cq(0,∞),q≥p≥0f \in {C^p}[0,\infty ) \cap {C^q}(0,\infty ),q \geq p \geq 0, andxif(p+i)(x)∈C[0,∞),i=1,…,q−p{x^i}{f^{(p + i)}}(x) \in C[0,\infty ),i = 1, \ldots ,q - p, whenq&gt;pq &gt; p. They appear for instance in certain Wiener-Hopf integral equations and are of interest if one wants to solve these by a Nyström method. To discretize the integral above, we propose to use a product rule of interpolatory type based on the zeros of Laguerre polynomials. For this rule we derive (weighted) uniform convergence estimates and present some numerical examples.

Convergence of Product Integration Rules Over (0, ∞) for Functions with Weak Singularities at the Origin

In this paper we consider integrals of the form ∫ 0 ∞ e −x K(x, y) f(x) dx, with f ∈ C p [0, ∞) ∩ C q (0, ∞), q ≥ p ≥ 0, and x i f (p+1) (x) ∈ C[0, ∞), i = 1,..., q − p, when q > p. They appear for instance in certain Wiener-Hopf integral equations and are of interest if one wants to solve these by a Nystrom method. To discretize the integral above, we propose to use a product rule of interpolatory type based on the zeros of Laguerre polynomials. For this rule we derive (weighted) uniform convergence estimates and present some numerical examples

Calculation of the Bethe logarithm for the Rydberg states of helium.

A very efficient technique for the finite-basis-set calculation of logarithmic sums is introduced. The basis sets contain sequences of nonlinear parameters determined by the zeros of Laguerre polynomials. It is shown that one can easily obtain convergence to 12 digits in fast calculations involving small basis sets. The method is then applied to calculations of the asymptotic expansion of the Bethe logarithm for the Rydberg states of helium. As perturbation calculations of the sums involved diverge, the present method is used to obtain accurate results for atoms in dipole and quadrupole fields from which the adiabatic contributions are extracted. A discussion of gauges in the multipole case and the suppression of roundoff errors by working in a mixed gauge is presented. A strategy for handling the nonadiabatic contributions to sums is presented and used to obtain preliminary bounds on these contributions.

Müntz systems and orthogonal Müntz-Legendre polynomials

The Müntz-Legendre polynomials arise by orthogonalizing the Müntz system { x λ 0 , x λ 1 , … } \{ {x^{{\lambda _0}}},{x^{{\lambda _1}}}, \ldots \} with respect to Lebesgue measure on [0, 1]. In this paper, differential and integral recurrence formulae for the Müntz-Legendre polynomials are obtained. Interlacing and lexicographical properties of their zeros are studied, and the smallest and largest zeros are universally estimated via the zeros of Laguerre polynomials. The uniform convergence of the Christoffel functions is proved equivalent to the nondenseness of the Müntz space on [0, 1], which implies that in this case the orthogonal Müntz-Legendre polynomials tend to 0 uniformly on closed subintervals of [0, 1). Some inequalities for Müntz polynomials are also investigated, most notably, a sharp L 2 {L^2} Markov inequality is proved.

On Appell-Laguerre polynomials

In this note we give a digest study of Appell-Laguerre polynomials, we provide a recurrence relation and a second-order differential equation satisfied by these polynomials. Moreover, an explicit expression and a generating function of the polynomials are given. The location and the number of the real zeros of Appell-Laguerre polynomials are determined; consequently, we recover information about the zeros of Laguerre polynomials when the range of the parameter α is the whole real axis excepting the negative integers.

On Extended Hermite-Fezér interpolation based on the zeros of Laguerre polynomials

Szasz [2] proved the existence of an interpolatory polynomial which interpolates a given function at given data with some extended points. He also gave its explicit representation. We study the uniform convergence of the same, interpolating at the zeros of Laguerre polynomials with one point (as origin) extension.

Relationship between equilibrium configurations of certain dynamical systems and zeros of Laguerre polynomials

Some dynamical systems coincide at equilibrium when the zeros of the generalized Laguerre polynomial satisfy certain nonlinear algebraic equations. The limiting cases for the zeros of Bessel functi...

On the algebra of certain matrices related to the zeros of Laguerre polynomials

On the algebra of certain matrices related to the zeros of Laguerre polynomials

Novel properties of the zeros of Laguerre polynomials

Novel properties of the zeros of Laguerre polynomials