Terms Of Whittaker Functions Research Articles

Analytic solution of Hedin’s equations in zero dimensions

Feynman diagrams for the many-body perturbational theory are enumerated by solving the system of Hedin’s equations in zero dimension. We extend the treatment of Molinari [Phys. Rev. B 71, 113102 (2005)] and give a complete solution of the enumeration problem in terms of Whittaker functions. An important relation between the generating function of the electron propagator and anomalous dimension in quantum field theory of massless fermions and mesons in four dimensions (Yukawa theory) is found. The Hopf algebra of undecorated rooted trees yields the anomalous field dimension in terms of the solution of the same differential equation. Its relation to the mathematical problem of combinatorics of chord diagrams is discussed; asymptotic expansions of the counting numbers are obtained.

Resonant states in an attractive one-dimensional cusp potential

We solve the two-component Dirac equation in the presence of a spatially one-dimensional symmetric attractive cusp potential. The components of the spinor solution are expressed in terms of Whittaker functions. We compute the bound states solutions and show that, as the potential amplitude increases, the lowest energy state sinks into the Dirac sea becoming a resonance. We characterize and compute the lifetime of the resonant state with the help of the phase shift and the Breit–Wigner relation. We discuss the limit when the cusp potential reduces to a delta point interaction.

Scattering of a relativistic scalar particle by a cusp potential

Abstract We solve the Klein–Gordon equation in the presence of a spatially one-dimensional cusp potential. The scattering solutions are obtained in terms of Whittaker functions and the condition for the existence of transmission resonances is derived. We show the dependence of the zero-reflection condition on the shape of the potential. In the low-momentum limit, transmission resonances are associated with half-bound states. We express the condition for transmission resonances in terms of the phase shifts.

A TUNABLE TRANSITION FROM FERROELECTRIC SLAB WAVEGUIDE TO MICROSTRIP, WITH APPLICATIONS TO RADAR TRANSMIT-RECEIVE SWITCHING

ABSTRACT A transmission-line model is used to obtain the reflection coefficient of a tapered transition from ferroelectric-slab waveguide to microstrip, both numerically and in terms of Whittaker functions. Such transitions are required due to the low impedances of ferroelectric slab waveguides, which are dictated by the need to bias these lines with dc fields. The model predicts that proper design can lead to transitions with excellent characteristics, even for strongly mismatched lines. It also predicts large changes in the reflection coefficient with an applied dc bias, which could make radar transmit-receive switching feasible.

Propagator and Slater sum in one‐body potential theory

AbstractFor a one‐body potential V(r) generating eigenfunctions ψi(r) and corresponding eigenvalues ϵi, the Feynman propagator K(r, r′, t) is simply related to the canonical density matrix C(r, r}′, β) by β → it. The diagonal element S(r, r},β) of C is the so‐called Slater sum of statistical mechanics. Differential equations for the Slater sum are first briefly reviewed, a quite general equation being available for a one‐dimensional potential V(x). This equation can be solved for a sech2 potential, and some physical properties of interest such as the local density of states are derived by way of illustration. Then, the Coulomb potential –Ze2/r is next considered, and it is shown that what is essentially the inverse Laplace transform of S(r, β)/β can be calculated for an arbitrary number of closed shells. Blinder has earlier determined the Feynman propagator in terms of Whittaker functions and contact is here established with his work. The currently topical case of Fermion vapours which are harmonically confined is then treated, for both two and three dimensions. Finally, in an Appendix, a perturbation series for the Slater sum is briefly summarized, to all orders in the one‐body potential V(r). The corresponding kinetic energy is thereby accessible.

On the integral of the workload process of the single server queue

This paper is devoted to a study of the integral of the workload process of the single server queue, in particular during one busy period. Firstly, we find asymptotics of the area 𝒜 swept under the workload process W(t) during the busy period when the service time distribution has a regularly varying tail. We also investigate the case of a light-tailed service time distribution. Secondly, we consider the problem of obtaining an explicit expression for the distribution of 𝒜. In the general GI/G/1 case, we use a sequential approximation to find the Laplace—Stieltjes transform of 𝒜. In the M/M/1 case, this transform is obtained explicitly in terms of Whittaker functions. Thirdly, we consider moments of 𝒜 in the GI/G/1 queue. Finally, we show asymptotic normality of .

On the integral of the workload process of the single server queue

This paper is devoted to a study of the integral of the workload process of the single server queue, in particular during one busy period. Firstly, we find asymptotics of the area 𝒜 swept under the workload process W(t) during the busy period when the service time distribution has a regularly varying tail. We also investigate the case of a light-tailed service time distribution. Secondly, we consider the problem of obtaining an explicit expression for the distribution of 𝒜. In the general GI/G/1 case, we use a sequential approximation to find the Laplace—Stieltjes transform of 𝒜. In the M/M/1 case, this transform is obtained explicitly in terms of Whittaker functions. Thirdly, we consider moments of 𝒜 in the GI/G/1 queue. Finally, we show asymptotic normality of .

Donor states in a multi-layered quantum dot

The ground and excited state energies of a hydrogenic impurity located at thecentre of a multi-layered quantum dot (MLQD) are calculated in the frameworkof the effective-mass approximation. The MLQD consists of a spherical core(e.g. GaAs) and a coated spherical shell (e.g. Ga1-xAlxAs). The wholedot is then embedded inside a bulk material (e.g. Ga1-yAlyAs). We solvethe Schrödinger equation exactly. The eigenfunctions of the impurity areexpressed in terms of Whittaker function and Coulomb wavefunction. The stateenergies are expressed in terms of the shell thickness, core radius, total dotradius and the potential heights. Our calculation shows that, as the dotradius approaches infinity, the state energies of an impurity located at thecentre of a multi-layered or a single-layered QD approach -1/n2 Ry,where n is the principal quantum number, Ry = µe4/2ε2ℏ2, µ and ε are the electronic effective mass and thedielectric constant of GaAs material. Thus it behaves like a three-dimensionalfree hydrogen atom. For very small dot radius, however, the state energy ofthe hydrogenic impurity of a MLQD behaves very differently from that of asingle-layered QD. For a multi-layered QD with finite shell and bulk potentialbarrier heights, the state energies of the impurity are found to be dependenton the difference of the shell potential (V2) and the bulk potential (V3).

Laplace transform of certain functions with applications

The Laplace transform of the functions tν(1+t)β, Reν > −1, is expressed in terms of Whittaker functions. This expression is exploited to evaluate infinite integrals involving products of Bessel functions, powers, exponentials, and Whittaker functions. Some special cases of the result are discussed. It is also demonstrated that the famous identity is a special case of our main result.

Inhomogeneous Electron Liquid: Differential Equation Satisfied by Diagonal Element of Coulomb Bound s-State Green Function

Because of density functional theory, attention has shifted from density matrices and Green functions to their diagonal elements. Here, an explicit partial differential equation is given for the diagonal elements of the bound state Green function for the s-states only. Relation to earlier work in terms of Whittaker functions is briefly discussed, for the Coulomb field case.

Neumann-Type Expansion of Coulomb Functions

An expansion is derived for the regular (power series) part of the Coulomb function, G0(η, ρ), in terms of Whittaker functions, which are closely related to the regular Coulomb functions F1 (η, ρ). The expansion coefficients are given as a sum of three terms; each of the terms obeys a simple three-term recurrence relation. In conjunction with the downward recurrence method for the regular functions (which is also discussed), this expansion is very useful for computing the irregular Coulomb functions G1(η, ρ), in particular for an attractive potential (η < 0) and for small or moderately large values of ρ

Landau-Zener transition to a decaying level.

Exact analytical solutions for the problem of term crossing are obtained for decaying levels with the help of the Laplace contour-integral method. For the case of fast passage through resonance and for the case of slow decay, simple asymptotic expressions are found from the exact solution in terms of Whittaker functions.

Hydrogenic impurity states in quantum dots and quantum wires.

The energies of hydrogenic impurity states with an impurity atom located at the center of a quantum dot and on the axis of a quantum-well wire are studied. These two systems are all assumed to have an infinite confining potential. In the case of the quantum dot, the impurity eigenfunctions are expressed in terms of Whittaker functions and Coulomb scattering functions. The calculated ground-state energy of the impurity approaches the correct limit of three-dimensional hydrogen atom as the radius of the quantum dot becomes very large. In the case of the quantum-well wire, analytical solutions can be obtained if we divide the space into a two-dimensional subspace (perpendicular to the axis of the quantum-well wire) and a one-dimensional subspace (parallel to the axis of the quantum-well wire). The calculated ground-state energy of the quantum-well wire approaches the ground-state energy of the shallow-impurity atom located on the surface as the radius of the wire becomes infinite. Variations of the state energies with the radius of the quantum dot and the quantum-well wire are obtained.

Raman free-electron laser with transverse density gradients.

The influence of the electron density gradients on the free-electron laser in the Raman regime is considered. Since in the Raman regime the electron density appears in the resonance parameter, the density gradients prevent the beam from being wholly resonant. The transverse gradients can then substantially reduce the gain even if the total current is kept constant. We solve a model problem of a free-electron laser, which employs a sheet electron beam in a planar wiggler, and which operates in the small-signal, high-gain regime. We study two cases, a density profile of uniform density gradient and a general density profile. For a uniform density gradient we write a dispersion relation explicitly in terms of Whittaker functions. For general density profiles, providing the current or the wiggler field is small, we derive an approximate dispersion relation through a solvability condition of the perturbed equations. In both cases, when the density gradient is large, the gain is expressed as a sum of contributions from the individual resonant layers, where each such contribution is inversely proportional to the density gradient at the respective resonant layer. A sufficient condition in order for the gain not to be substantially reduced by the density gradients is formulated.

An initial boundary-value problem for the Schrödinger equation with long-range potential

We consider the exterior Dirichlet problem in R 3 for the Schrödinger equation with a Coulomb potential. For such a Schrödinger equation defined in all of R 3 , the so-called reference problem, a fundamental solution is known in terms of Whittaker functions and the appropriate radiation conditions and the asymptotic behaviour of solutions can easily be obtained. Consequently, we can prove the limiting absorption principle, establish the existence of solutions to exterior boundary-value problems and show that the associated wave operators exist. This then enables questions concerning the asymptotic behaviour of time depen­dent solution of the exterior problem to be resolved in terms of the reference problem.

On a Class of Electromagnetic Wave Functions for Propagation Along the Circular Gyrotropic Waveguide

The properties of confluent hypergeometric functions as exact electromagnetic wave fnnctions for propagation in a circular waveguide containing azimuthally magnetized remanent ferrite are investigated. Two different forms of solutions of the propagation problem for angular symmetric transverse electric modes are constructed-one in terms of Kummer and Tricomi confluent hypergeometric functions of complex parameter and variable and a second in terms of Whittaker functions. An evaluation of this class of wave functions is performed to sufficient extent, followed by tabulation of their imaginary zeros, providing computation of eigenvalue spectrum and phase characteristics of the gyrotropic guide.

The Effect of Current Shear on Topographic Rossby Waves

Abstract Topographic Rossby waves are long-period waves which occur on continental slopes. In this paper we examine the effect of a mean current on these waves, when the current is directed along the isobaths and has a linear shear in the transverse direction. Solutions are obtained in terms of Whittaker functions and are applied to some data analyzed by Hamon et al. (1975), which has been interpreted as evidence of topographic Rossby waves by Garrett (1979). The modifications due to a linear friction law are also considered.

Characterization of acoustic disturbances in linearly sheared flows

The equation describing the plane wave propagation, the stability, or the rectangular duct mode characteristics in a compressible inviscid linearly sheared parallel, but otherwise homogeneous, flow, is shown to be reducible to Whittaker's equation. The resulting solutions, which are real, viewed as functions of two variables, depend on a parameter and an argument the values of which have precise physical meanings depending on the problem. The exact solutions in terms of Whittaker functions are used to obtain a number of known results of plane wave propagation and stability in linearly sheared flows as limiting cases in which the speed of sound goes to infinity (incompressible limit) or the shear layer thickness, or wave number, goes to zero (vortex sheet limit). The usefulness of the exact solutions is then discussed in connection with the problems of plane wave propagation and stability of a finite thickness shear layer with a linear velocity profile. With respect to the plane wave propagation it is shown that, unlike the compressible vortex sheet, the shear layer possesses no resonances and no Brewster angles, whereas with respect to the stability problem it is shown that, again unlike the compressible vortex sheet, the thin layer is unstable to long wavelength disturbances for all Mach numbers. These results imply that the reflection and stability characteristics of a non-zero thickness but thin shear layer (i.e., the long wavelength characteristics) do not go over smoothly into the results of the compressible vortex sheet as the wave number approaches zero, except for a limited range of generally subsonic relative flow of the two parallel streams bounding the shear layer.

Radial deformation of a nonhomogeneous spherically isotropic elastic sphere with a concentric spherical inclusion

The elasticity of a spherically isotropic medium bounded by two concentric spherical surfaces subjected to normal pressures is discussed. The material of the structure is spherically isotropic and, in addition, is continuously inhomogeneous with mechanical properties varying exponentially as the square of the radius. An exact solution of the problem in terms of Whittaker functions is presented. The St. Venant’s solution in the case of homogeneous material and Lame’s solution in the case of homogeneous isotropic material are derived from the general solution. The problem of a solid sphere of the same medium under the external pressure is also solved as a particular case of the above problem. Finally, the displacements and stresses of a composite sphere consisting of a solid spherical body made of homogeneous material and a nonhomogeneous concentric spherical shell covering the inclusion, both of them being spherically isotropic, are obtained when the sphere is under uniform compression.

The Klein-Gordon equation in a Kerr-Newman background space

The time-dependent Klein-Gordon equation for a massive scalar meson field is examined in the Kerr-Newman background space and solutions near the outer horizon and at infinity are obtained in terms of Whittaker functions. In the special case of a Schwarzschild black hole the solutions are shown to be closely related to asymptotic solutions derived elsewhere.