Parabolic Cylinder Functions Research Articles

Some integral representations and limits for (products of) the parabolic cylinder function

ABSTRACTRecently, [Veestraeten D. On the inverse transform of Laplace transforms that contain (products of) the parabolic cylinder function. Integr Transf Spec F 2015;26:859–871] derived inverse Laplace transforms for Laplace transforms that contain products of two parabolic cylinder functions by exploiting the link between the parabolic cylinder function and the transition density and distribution functions of the Ornstein–Uhlenbeck process. This paper first uses these results to derive new integral representations for (products of two) parabolic cylinder functions. Second, as the Brownian motion process with drift is a limiting case of the Ornstein–Uhlenbeck process also limits can be calculated for the product of gamma functions and (products of) parabolic cylinder functions. The central results in both cases contain, in stylized form, and such that the recurrence relation of the parabolic cylinder function straightforwardly allows to obtain integral representations and limits also for countless other combinations in the orders such as and.

On the inverse transform of Laplace transforms that contain (products of) the parabolic cylinder function

The Laplace transforms of the transition probability density and distribution functions for the Ornstein–Uhlenbeck process contain the product of two parabolic cylinder functions, namely and , respectively. The inverse transforms of these products have as yet not been documented. However, the transition density and distribution functions can be obtained by alternatively applying Doob's transform to the Kolmogorov equation and casting the problem in terms of Brownian motion. Linking the resulting transition density and distribution functions to their Laplace transforms then specifies the inverse transforms to the aforementioned products of parabolic cylinder functions. These two results, the recurrence relation of the parabolic cylinder function and the properties of the Laplace transform then enable the calculation of inverse transforms also for countless other combinations in the orders of the parabolic cylinder functions such as , and .

An integral representation for the product of two parabolic cylinder functions having unrelated arguments

An integral representation is provided for the parabolic cylinder function product , where and are unrelated. A few simple consequences are given in the form of hyperbolic integrals and a sum rule.

Refined error estimates for the Riccati equation with applications to the angular Teukolsky equation

We derive refined rigorous error estimates for approximate solutions of Sturm-Liouville and Riccati equations with real or complex potentials. The approximate solutions include WKB approximations, Airy and parabolic cylinder functions, and certain Bessel functions. Our estimates are applied to solutions of the angular Teukolsky equation with a complex aspherical parameter in a rotating black hole Kerr geometry.

Controllable parabolic-cylinder optical rogue wave.

We demonstrate controllable parabolic-cylinder optical rogue waves in certain inhomogeneous media. An analytical rogue wave solution of the generalized nonlinear Schrödinger equation with spatially modulated coefficients and an external potential in the form of modulated quadratic potential is obtained by the similarity transformation. Numerical simulations are performed for comparison with the analytical solutions and to confirm the stability of the rogue wave solution obtained. These optical rogue waves are built by the products of parabolic-cylinder functions and the basic rogue wave solution of the standard nonlinear Schrödinger equation. Such rogue waves may appear in different forms, as the hump and paw profiles.

Breather solutions of the generalized nonlinear Schrödinger equation with spatially modulated parameters and a special external potential

By applying the similarity transformation we find analytical solutions of the generalized nonlinear Schrodinger equation with spatially modulated parameters and a special external potential in the form of periodic breathing nonlinear wave packets. These breathers are built by the products of parabolic-cylinder functions and the one-soliton solutions with the continuous-wave background or the second-order soliton solutions of the standard nonlinear Schrodinger equation with constant coefficients. Some examples of such composed solutions possessing applicative potential are given, in which the breathers appear in different forms. The numerical simulations are performed to compare the numerical solution with the analytical solution and to confirm the stability of the breathers found.

Persistence of random walk records

We study records generated by Brownian particles in one dimension. Specifically, we investigate an ordinary random walk and define the record as the maximal position of the walk. We compare the record of an individual random walk with the mean record, obtained as an average over infinitely many realizations. We term the walk ‘superior’ if the record is always above average, and conversely, the walk is said to be ‘inferior’ if the record is always below average. We find that the fraction of superior walks, S, decays algebraically with time, S ∼ t−β, in the limit t → ∞, and that the persistence exponent is nontrivial, β = 0.382 258…. The fraction of inferior walks, I, also decays as a power law, I ∼ t−α, but the persistence exponent is smaller, α = 0.241 608…. Both exponents are roots of transcendental equations involving the parabolic cylinder function. To obtain these theoretical results, we analyze the joint density of superior walks with a given record and position, while for inferior walks it suffices to study the density as a function of position.

The Relationship Between Semiclassical Laguerre Polynomials and the Fourth Painlevé Equation

We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semi-classical Laguerre weight and classical solutions of the fourth Painlev\'e equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions which arise in the description of special function solutions of the fourth Painlev\'e equation.

Edge states in bilayer graphene in a magnetic field

Edge states in biased bilayer graphene in a magnetic field are studied within the four-band continuum model. The analysis is done for the semi-infinite graphene plane and for the graphene ribbon of a finite width, in the cases of zigzag and armchair edges. Exact dispersion equations for the edge states and analytic expressions for their wave functions are written in terms of the parabolic cylinder functions. The spectrum of edge states for each type of the boundary conditions is found by numerically solving the corresponding dispersion equations. The low-energy modes localized at zigzag edges are explored in detail.

ON THE MATHIEU EQUATION DESCRIBING ULTRA-RELATIVISTIC FERMIONS EVOLVING IN A TIME-HARMONIC ELECTRIC FIELD PARALLEL TO A STATIC MAGNETIC INDUCTION

The present work is studying the behavior of relativistic fermions in parallel electric and magnetic fields, oriented along Oz. As expected, in the particular case of static field intensities, the temporal part of the wave function is expressed in terms of the parabolic cylinder functions. When the static electric field is replaced by a time-harmonic one, the ultra-relativistic particles are described by a Mathieu-type equation, of purely imaginary parameter. Below the first branching point, which imposes a maximum value for the electric intensity, one may use a series expansion of the Mathieu's functions to derive the corresponding wave functions and the conserved current density components. Finally, we obtain a nontrivial quantization law of the longitudinal momentum, pz.

High-index asymptotics of spherical Bessel products averaged with modulated Gaussian power laws

Bessel integrals of type \({\int_0^\infty {g(k)j_l^{(m)}(k)j_l^{(n)} (k)k^{2}{\rm d}k}}\) are investigated, where the kernel g(k) is a modulated Gaussian power-law distribution \({k^\mu{e}^{-ak^{2}-(b+{{\rm i}} \omega)k}}\), and the jl(m) are multiple derivatives of spherical Bessel functions. These integrals define the multipole moments of Gaussian random fields on the unit sphere, arising in multipole fits of temperature and polarization power spectra of the cosmic microwave background. Two methods allowing efficient numerical calculation of these integrals are presented, covering Bessel indices l in the currently accessible multipole range 0 ≤ l ≤ 104 and beyond. The first method is based on a representation of spherical Bessel functions by Lommel polynomials. Gaussian power-law averages can then be calculated in closed form as finite Hankel series of parabolic cylinder functions, which allow high-precision evaluation. The second method is asymptotic, covering the high-l regime, and is applicable to general distribution functions g(k) in the integrand; it is based on the uniform Nicholson approximation of the Bessel derivatives in conjunction with an integral representation of squared Airy functions. A numerical comparison of these two methods is performed, employing Gaussian power laws and Kummer distributions to average the Bessel products.

Conical functions of purely imaginary order and argument

Associated Legendre functions are studied for the case where the degree is in conical form −½ + iτ (τ real), and the order iμ and argument ix are purely imaginary (μ and x real). Conical functions in this form have applications to Fourier expansions of the eigenfunctions on a closed geodesic. Real-valued numerically satisfactory solutions are introduced which are continuous for all real x. Uniform asymptotic approximations and expansions are then derived for the cases where one or both of μ and τ are large; these results (which involve elementary, Airy, Bessel and parabolic cylinder functions) are uniformly valid for unbounded x.

Differential commuting operator and closed-form eigenfunctions for linear canonical transforms

The linear canonical transform (LCT) with a, b, c, d parameter plays an important role in quantum mechanics, optics, and signal processing. The eigenfunctions of the LCT are also important because they describe the self-imaging phenomenon in optical systems. However, the existing solutions for the eigenfunctions of the LCT are divided into many cases and they lack a systematic way to solve these eigenfunctions. In this paper, we find a linear, second-order, self-adjoint differential commuting operator that commutes with the LCT operator. Hence, the commuting operator and the LCT share the same eigenfunctions with different eigenvalues. The commuting operator is very general and simple when it is compared to the existing multiple-parameter differential equations. Then, the eigenfunctions can be derived systematically. The eigenvalues of the commuting operator have closed-form relationships with the eigenvalues of the LCT. We also simplify the eigenfunctions for |a+d|>2 and a+d=±2, b≠0 into the more compact closed form instead of the integral form. For |a+d|>2, the eigenfunctions are related to the parabolic cylinder functions.

Existence results for nano boundary layer flows with nonlinear Navier boundary condition

The standard no slip boundary condition of classical fluid mechanics is no longer valid at the micro- and nano-scale and should be replaced by a boundary condition that allows some degree of tangential slip. In the present work, the classical laminar boundary layer equation of the flow away from the origin past a wedge with the no-slip boundary condition replaced by a nonlinear Navier boundary condition is revisited. This boundary condition includes an arbitrary index parameter, denoted by n > 0 , which appears in the coefficients of the differential equation to be solved. It is proved corresponding to the value n = 1 3 , there are exactly three situations for the problem: (i) there is no solution; (ii) there exist two solutions; (iii) there exist four solutions. Furthermore, the exact analytical solution of the problem is given in terms of parabolic cylinder functions for further physical interpretations.

On three methods for analytic Laplace inversion in the framework of Brownian motion and their excursions

Working in a framework originating with Brownian motion and its excursions, this paper establishes a two-step Laplace inversion method for determining a function which is known through its transform after a convolution with another function with a known transform. The first step here has as its domain the class of parabolic cylinder functions, and it develops analytic Laplace inversion of their reciprocals. The second step pertains to convolutions on the positive reals with analytic factors where one of them is of exponential-order decay to zero at the origin; it develops two Laplace-inversion-based methods for handling these by asymptotic expansions. The results are shown to have applications to finance, yielding series representations and asymptotic expansions for the valuation and hedging of Parisian barrier options.

Accessible spatiotemporal parabolic-cylinder solitons

We study analytically and numerically ‘accessible’ spatiotemporal solitons in a three-dimensional strongly nonlocal nonlinear medium. A general localized soliton solution of the ‘acceptable’ type is obtained in the Cartesian coordinates, using even and odd parabolic-cylinder functions. Characteristics of these accessible spatiotemporal solitons are discussed. The validity of the analytical solutions and their stability is verified by means of direct numerical simulations.

Computing the complex zeros of special functions

The complex zeros of solutions of second order linear ODEs lie over certain curves in the complex plane called anti-Stokes lines. We consider the Liouville-Green (WKB) approximation for linear homogeneous second order ODEs $$w^{\prime \prime }(z)+A(z)w(z)=0$$ with $$A(z)$$ meromorphic, and describe the qualitative properties of the approximate anti-Stokes lines. Based on this qualitative description, we construct a fourth order method which efficiently computes the complex zeros of solutions of second order ODEs following the path of the approximate anti-Stokes lines. We illustrate the method with the computation of the zeros of parabolic cylinder functions, Bessel functions and generalized Bessel polynomials and describe specific algorithms for the computation of these zeros.

Global Asymptotics for Meixner‐Pollaczek Polynomials with a Varying Parameter

In this paper, we study the uniform asymptotics of the Meixner‐Pollaczek polynomials with varying parameter as , where A > 0 is a constant. Two asymptotic expansions are obtained, which hold uniformly for z in two overlapping regions which together cover the whole complex plane. One involves parabolic cylinder functions, and the other is in terms of elementary functions only. Our approach is based on the steepest descent method for oscillatory Riemann‐Hilbert problems first introduced by Deift and Zhou [1].

Singular Isotonic Oscillator, Supersymmetry and Superintegrability

In the case of a one-dimensional nonsingular Hamiltonian $H$ and a singular supersymmetric partner $H_a$, the Darboux and factorization relations of supersymmetric quantum mechanics can be only formal relations. It was shown how we can construct an adequate partner by using infinite barriers placed where are located the singularities on the real axis and recover isospectrality. This method was applied to superpartners of the harmonic oscillator with one singularity. In this paper, we apply this method to the singular isotonic oscillator with two singularities on the real axis. We also applied these results to four 2D superintegrable systems with second and third-order integrals of motion obtained by Gravel for which polynomial algebras approach does not allow to obtain the energy spectrum of square integrable wavefunctions. We obtain solutions involving parabolic cylinder functions.

Turán Type Inequalities for Tricomi Confluent Hypergeometric Functions

Some sharp two-sided Turan type inequalities for parabolic cylinder functions and Tricomi confluent hypergeometric functions are deduced. The proofs are based on integral representations for quotients of parabolic cylinder functions and Tricomi confluent hypergeometric functions, which arise in the study of the infinite divisibility of the Fisher–Snedecor F distribution. Moreover, some complete monotonicity results are given concerning Turan determinants of Tricomi confluent hypergeometric functions. These complement and improve some of the results of Ismail and Laforgia (in Constr. Approx. 26:1–9, 2007).