Terms Of Whittaker Functions Research Articles

Study of the Klein–Gordon equation for a hydrogenic model of dyons

This paper presents the generalization of a zero spin hydrogen atom to a relativistic atomic model of hydrogen with dyons using the Klein–Gordon equation. The derivation of the Klein–Gordon equation for the particle of relative motion is shown. In addition, the analytical solutions of the equation are calculated in terms of Whittaker functions and Jacobi weighted polynomials. The discrete spectrum of energy and the charge density of the orbiting dyon are presented. For a system of positive magnetic and electric charges in the nucleus and negative charges for the orbiting particle, and considering the first allowed values of [Formula: see text] and [Formula: see text], it was found that the dyon atom acts with a greater force of interaction between the charges of the nucleus and the secondary particle compared to the standard atom. It was obtained by comparing the distance between the nucleus and charge density concentrations from the dyon atom with the relativistic pionic atom.

Holomorphic Family of Dirac–Coulomb Hamiltonians in Arbitrary Dimension

AbstractWe study massless one-dimensional Dirac–Coulomb Hamiltonians, that is, operators on the half-line of the form $$D_{\omega ,\lambda }:=\begin{bmatrix} -\frac{\lambda +\omega }{x} &{} - \partial _x \\ \partial _x &{} -\frac{\lambda -\omega }{x} \end{bmatrix}$$ D ω , λ : = - λ + ω x - ∂ x ∂ x - λ - ω x . We describe their closed realizations in the sense of the Hilbert space $$L^2(\mathbb {R}_+,\mathbb {C}^2)$$ L 2 ( R + , C 2 ) , allowing for complex values of the parameters $$\lambda ,\omega $$ λ , ω . In physical situations, $$\lambda $$ λ is proportional to the electric charge and $$\omega $$ ω is related to the angular momentum. We focus on realizations of $$D_{\omega ,\lambda }$$ D ω , λ homogeneous of degree $$-1$$ - 1 . They can be organized in a single holomorphic family of closed operators parametrized by a certain two-dimensional complex manifold. We describe the spectrum and the numerical range of these realizations. We give an explicit formula for the integral kernel of their resolvent in terms of Whittaker functions. We also describe their stationary scattering theory, providing formulas for a natural pair of diagonalizing operators and for the scattering operator. We describe the point spectrum of their nonhomogeneous realizations. It is well-known that $$D_{\omega ,\lambda }$$ D ω , λ arise after separation of variables of the Dirac–Coulomb operator in dimension 3. We give a simple argument why this is still true in any dimension. Furthermore, we explain the relationship of spherically symmetric Dirac operators with the Dirac operator on the sphere and its eigenproblem. Our work is mainly motivated by a large literature devoted to distinguished self-adjoint realizations of Dirac–Coulomb Hamiltonians. We show that these realizations arise naturally if the holomorphy is taken as the guiding principle. Furthermore, they are infrared attractive fixed points of the scaling action. Beside applications in relativistic quantum mechanics, Dirac–Coulomb Hamiltonians are argued to provide a natural setting for the study of Whittaker (or, equivalently, confluent hypergeometric) functions.

Extreme dilaton black holes in 2 + 1 dimensions: quasinormal modes

We study neutral massless scalar field perturbations around an extreme dilaton black hole in 2 + 1 dimensions: the wave equations of the massless scalar field is shown to be exactly solvable in terms of Whittaker functions. Thus, the quasinormal modes are computed exactly and shown to be purely imaginary: we show the existence of stable and unstable modes. Interestingly, the quasinormal modes do not depend on the black holes parameters and the fundamental mode is always unstable and depends only on the parameters of the test field. Also, we determine the quasinormal frequencies via the improved asymptotic iteration method which shows a good agreement with the analytical results.

Analytical Solutions of the Internal Gravity Wave Equation for a Semi-Infinite Stratified Layer of Variable Buoyancy

The problem of constructing asymptotics describing far-field internal gravity waves generated by an oscillating point source of perturbations moving in a vertically semi-infinite stratified layer of variable buoyancy is considered. For a model distribution of the buoyancy frequency, analytical solutions of the main boundary value problem are obtained, which are expressed in terms of Whittaker functions. An integral representation for the Green’s function is obtained, and asymptotic solutions are constructed that describe the amplitude-phase characteristics of internal gravity wave fields in a semi-infinite stratified medium with a variable buoyancy frequency far away from the perturbation source.

Love wave frequency in an orthotropic crust over a double-layered anisotropic mantle

In the present paper, investigation has been made to study the propagation character of Love wave in an orthotropic substratum being layered doubly over anisotropic porous mantle. Separate expressions for displacement components have been derived for each of the three layer and suitable boundary conditions have been imposed to derive the dispersion equation of the wave in a closed form in terms of Whittaker function. It has been found that inhomogeneity parameter associated with the rigidity and the density of the layered medium has a special bearing on the Love wave propagation. Orthotropic properties and initial stresses of the media also affects the velocity of the wave to a great extent has been shown graphically.

Microstructure and inertial characteristics of a magnetite ferrofluid over a stretching/shrinking sheet using effective thermal conductivity model

Nanofluid is the most promising gift of modern science to improve the heat transfer capabilities of conventional heat transfer fluids. However, one of the most crucial drawbacks for classical nanofluid models is that they cannot describe a class of fluids that have certain microscopic characters arising from the micro-rotation and local structure of the fluid elements. In this work, the innovative micropolar nanofluid model is introduced to study the microstructure and inertial characteristics of the substructure particles. More exactly, the flow and heat transport of micropolar ferrofluid over a stretching/shrinking sheet subjected to suction and injection is studied. Magnetite-Fe3O4 (iron oxide) nanoparticles are considered in water taken as conventional base fluid. The mathematical model has been formulated based on Tiwari-Das nanofluid model. Explicit exact solutions of non-linear coupled momentum equations are obtained. The solution of energy equation is obtained in terms of Whittaker function with the help of Maple. The impacts of pertinent parameters on velocity, micro-rotation velocity and temperature are shown graphically for positive and negative mass transfer flow and analyzed in detail. The results show that micro-rotation velocity increases first and then decreases. There is a remarkable change occurs to micro-rotation velocity for positive and negative values of mass transfer parameter. Presence of mass transfer parameter accelerate the profiles near the flow domain and then decelerates it. Further, micropolar ferrofluid have higher velocity than the classical nanofluid. Comparison have been made with published data under special cases and obtained in close agreement.

Laminar counterflow parallel-plate heat exchangers: An exact solution including axial and transverse wall conduction effects

Multilayered, counterflow, parallel-plate heat exchangers are studied theoretically and numerically. The analysis, carried out for constant property fluids, assumes a fully developed laminar flow with moderately large Peclet numbers in the flow channels, so that longitudinal conduction can be neglected in the fluids in first approximation. By way of contrast, both axial and transverse conduction effects are fully accounted for in the heat conducting plates, making the analysis relevant to the design of mini- and micro-heat exchangers. The exact solution for the temperature field is obtained in terms of eigenfunction expansions involving infinite sets of both positive and negative eigenvalues. Based on previous results, the eigenfunctions are expressed in terms of Whittaker functions, leading to an analytical eigencondition that provides the eigenvalues numerically. Making use of a newly defined orthogonality condition, the expansion coefficients are determined through an infinite system of linear equations that must be truncated to a sufficiently large number of terms to obtain a numerical solution of the problem. The exact solution for the temperature field provides analytical expressions for the interfacial and bulk temperatures, local heat-transfer rates, overall heat-transfer coefficient, Nusselt numbers, and outlet bulk temperatures of both fluids. Numerical results evaluated using a moderate number of terms in the series are compared against finite difference solutions of the problem and against previous results reported in the literature, showing excellent agreement in both cases. The exact solution presented here may be used as a benchmark case for computational heat transfer codes, and enables future spectral and parametric studies of the problem.

Influence of initial stress and gravity on torsional surface wave in heterogeneous medium

The present paper investigates the effect of gravity and initial stress on the propagation of torsional surface waves in heterogeneous medium. The dispersion equation has been obtained for rigid and traction free boundaries in terms of Whittaker function. The present study reveals that torsional surface waves can propagate in both the cases. In this paper, we assume the expansion of Whittaker function up to linear term. It is observed that, in presence of initial stress and gravity for both rigid and traction free boundary, the phase velocity of torsional surface waves increases with the growth of rigidity. In both the cases, it has also been noticed that as the gravity increases, the phase velocity of the torsional surface waves decreases in presence of heterogeneity and initial stress. It is concluded that in presence of traction free boundary, the phase velocity of the torsional surface waves is more than the rigid boundary.

SH-wave velocity in a fiber-reinforced anisotropic layer overlying a gravitational heterogeneous half-space

Purpose – The purpose of this paper is to investigate the existence of SH-waves in fiber-reinforced layer placed over a heterogeneous elastic half-space. Design/methodology/approach – The heterogeneity of the elastic half-space is caused by the exponential variations of density and rigidity. As a special case when both the layers are homogeneous, the derived equation is in agreement with the general equation of Love wave. Findings – Numerically, it is observed that the velocity of SH-waves decreases with the increase of heterogeneity and reinforced parameters. The dimensionless phase velocity of SH-waves increases with the decreases of dimensionless wave number and shown through figures. Originality/value – In this work, SH-wave in a fiber-reinforced anisotropic medium overlying a heterogeneous gravitational half-space has been investigated analytically and numerically. The dispersion equation for the propagation of SH-waves has been observed in terms of Whittaker function and its derivative of second degree order. It has been observed that on the removal of heterogeneity of half-space, and reinforced parameters of the layer, the derived dispersion equation reduces to Love wave dispersion equation thereby validates the solution of the problem. The equation of propagation of Love wave in fiber-reinforced medium over a heterogeneous half-space given by relevant authors is also reduced from the obtained dispersion relation under the considered geometry.

Scattering states of Cusp potential in minimal length Dirac equation

In this article, we investigate the scattering state of spin-half particle within the framework of minimal length quantum mechanics for Cusp potential in terms of Whittaker functions. We calculate the reflection and transmission coefficients for spin and pseudospin symmetry limits using the conditions of equation of continuity of the wave functions.

Tracy-Widom asymptotics for a random polymer model with gamma-distributed weights

We establish Tracy-Widom asymptotics for the partition function of a random polymer model with gamma-distributed weights recently introduced by Seppalainen. We show that the partition function of this random polymer can be represented within the framework of the geometric RSK correspondence and consequently its law can be expressed in terms of Whittaker functions. This leads to a representation of the law of the partition function which is amenable to asymptotic analysis. In this model, the partition function plays a role analogous to the smallest eigenvalue in the Laguerre unitary ensemble of random matrix theory.

Optical properties of hydrogenic impurity in an inhomogeneous infinite spherical quantum dot

In the present work, using the effective mass approximation, the Schrödinger equation of system is solved in terms of Whittaker functions. The linear and third-order nonlinear optical absorption coefficient (AC) as well as refractive index (RI) changes associated with two intersubbund transitions (1s–2p and 2p–3d) in the case of a GaAs inhomogeneous infinite spherical quantum dot are investigated at different inner radii of shell and shell thicknesses. Regarding this, the optical properties of hydrogenic system are studied by means of compact density approach and dipole approximation. The results show that the system under study is strongly affected by inner radius of shell and shell thickness changes. Also it was found that the transition between orbital with bigger l value shift to higher photon energy region.

Study of oscillator strengths of hydrogenic impurity in a inhomogeneous finite and infinite spherical quantum dots

In this study, using the effective mass approximation and continuum dielectric media, the Schrödinger equation of nano systems is solved in terms of Whittaker functions. The oscillator strengths of both finite and infinite confining potentials from ground to the first and second allowed excited states are calculated with hydrogenic impurity. Regarding this, the oscillator strengths of system with hydrogenic impurity are studied by means of dipole approximation. The results show that in the case of constant shell thickness, when the inner radius of shell thickness is small, the value of oscillator strengths of both transitions (from ground to the first and second allowed excited states) is strongly affected by changes in shell thickness. In the large enough inner radius of shell thickness, the corresponding oscillator strength curves coincide for two transitions. Generally, these two transitions behave differently, i.e. by increasing the value of inner radius of shell, the first two transitions decrease and decrease respectively (due to conservation of probability).

Geometric RSK correspondence, Whittaker functions and symmetrized random polymers

We show that the geometric lifting of the RSK correspondence introduced by A.N. Kirillov (Physics and Combinatorics. Proc. Nagoya 2000 2nd Internat Workshop, pp. 82–150, 2001) is volume preserving with respect to a natural product measure on its domain, and that the integrand in Givental’s integral formula for $\mathit{GL}(n,{\mathbb{R}})$ -Whittaker functions arises naturally in this context. Apart from providing further evidence that Whittaker functions are the natural analogue of Schur polynomials in this setting, our results also provide a new ‘combinatorial’ framework for the study of random polymers. When the input matrix consists of random inverse gamma distributed weights, the probability distribution of a polymer partition function constructed from these weights can be written down explicitly in terms of Whittaker functions. Next we restrict the geometric RSK mapping to symmetric matrices and show that the volume preserving property continues to hold. We determine the probability law of the polymer partition function with inverse gamma weights that are constrained to be symmetric about the main diagonal, with an additional factor on the main diagonal. The third combinatorial mapping studied is a variant of the geometric RSK mapping for triangular arrays, which is again showed to be volume preserving. This leads to a formula for the probability distribution of a polymer model whose paths are constrained to stay below the diagonal. We also show that the analogues of the Cauchy-Littlewood identity in the setting of this paper are equivalent to a collection of Whittaker integral identities conjectured by Bump (Number Theory, Trace Formulas, and Discrete Groups, pp. 49–109, 1989) and Bump and Friedberg (Festschrift in Honor of Piatetski-Shapiro, Part II, pp. 47–65, 1990) and proved by Stade (Am. J. Math. 123:121–161, 2001; Israel J. Math. 127:201–219, 2002). Our approach leads to new ‘combinatorial’ proofs and generalizations of these identities, with some restrictions on the parameters.

Dispersion of torsional surface waves in anisotropic layer over porous half space under gravity

AbstractIn this paper an attempt has been made to derive the frequency equation of torsional surface waves in initially stressed fiber reinforced layer lying over fluid saturated anisotropic porous half space under gravity. The dispersion equation has been obtained in terms of Whittaker function and its derivatives, which further expanded asymptotically and retain the terms upto second degree. The analysis of frequency equation bear out the pronounced effect of initial stress, transverse and longitudinal rigidity of reinforced material, gravity and porosity of the porous half space on the propagation of torsional surface waves. Moreover, it can be noted that in the absence of initial stress, reinforcement parameters, gravity and porosity, the frequency equation of torsional surface waves reduces to dispersion equation of Love waves in isotropic layer. This ensures the classical result that torsional surface waves do not propagate through isotropic homogeneous medium.

Love waves in the fiber-reinforced layer over a gravitating porous half-space

This paper aims to study the propagation of Love waves in fiber-reinforced layer lying over a gravitating anisotropic porous half-space. The closed form of dispersion equation has been derived for the Love waves in terms of Whittaker function and its derivative, which are further expanded asymptotically, retaining the terms up to second degree. The frequency equation shows that the transverse and longitudinal rigidity of reinforced material, as well as gravity and porosity of the porous halfspace have significant effect on the propagation of Love waves. The study reveals that the increment in width of reinforced layer decreases the phase velocity. For a particular width of the reinforced layer, it is also observed that the phase velocity increases with increasing porosity of the half-space, but it decreases with increasing gravity.

Glueball masses from linearly confining supergravity

Mass spectrum of 0++ glueballs is produced using a dual supergravity theory we proposed for pure N=1 SU(N) gauge theory in four dimensions in the large N limit in the IR. The glueball states are expressed in terms of Whittaker functions. The spectrum is discrete and a function whose roots give the masses is written. The ratios of the masses are independent of the parameters of the theory and comparison to recent non-supersymmetric large N lattice QCD data available for the lowest three states shows agreement to within five percent.

Laminar counterflow parallel-plate heat exchangers: Exact and approximate solutions

Multilayered, counterflow, parallel-plate heat exchangers are analyzed numerically and theoretically. The analysis, carried out for constant property fluids, considers a hydrodynamically developed laminar flow and neglects longitudinal conduction both in the fluid and in the plates. The solution for the temperature field involves eigenfunction expansions that can be solved in terms of Whittaker functions using standard symbolic algebra packages, leading to analytical expressions that provide the eigenvalues numerically. It is seen that the approximate solution obtained by retaining the first two modes in the eigenfunction expansion provides an accurate representation for the temperature away from the entrance regions, specially for long heat exchangers, thereby enabling simplified expressions for the wall and bulk temperatures, local heat-transfer rate, overall heat-transfer coefficient, and outlet bulk temperatures. The agreement between the numerical and theoretical results suggests the possibility of using the analytical solutions presented herein as benchmark problems for computational heat-transfer codes.

On minimum Fisher information distributions with restricted support and fixed variance

Fisher information is of key importance in estimation theory. It also serves in inference problems as well as in the interpretation of many physical processes. The mean-squared estimation error for the location parameter of a distribution is bounded by the inverse of the Fisher information associated with this distribution. In this paper we look for minimum Fisher information distributions with a restricted support. More precisely, we study the problem of minimizing the Fisher information in the set of distributions with fixed variance defined on a bounded subset S of R or on the positive real line. We show that the solutions of the underlying differential equation can be expressed in terms of Whittaker functions. Then, in the two considered cases, we derive the explicit expressions of the solutions and investigate their behavior. We also characterize the behavior of the minimum Fisher information as a function of the imposed variance.

The nearly horizontally rolling of a thick disk on a rough plane

In the paper the analytical solution of the partially linearized equations of motion of nearly horizontally rolling of a thick rigid disk on a perfectly rough horizontal plane under the action of gravity is given in terms of Whittaker functions. The solution is used to obtain the asymptotic solutions for a very small inclination angle, the study of unilateral contact between a disk and a plane and the study of disk colliding motion.