Spheroidal Wave Equation Research Articles

Computation of the eigenvalues for the angular and Coulomb spheroidal wave equation

In this paper we study the eigenvalues of the angular spheroidal wave equation and its generalization, the Coulomb spheroidal wave equation. An associated differential system and a formula for the connection coefficients between the various Floquet solutions give rise to an entire function whose zeros are exactly the eigenvalues of the Coulomb spheroidal wave equation. This entire function can be calculated by means of a recurrence formula with arbitrary accuracy and low computational cost. In the end, one obtains an easy-to-use method for computing spheroidal eigenvalues and spheroidal wave functions.

The Harmonic Lagrange Top and the Confluent Heun Equation

The harmonic Lagrange top is the Lagrange top plus a quadratic (harmonic) potential term. We describe the top in the space fixed frame using a global description with a Poisson structure on \(T^{*}S^{3}\). This global description naturally leads to a rational parametrisation of the set of critical values of the energy-momentum map. We show that there are 4 different topological types for generic parameter values. The quantum mechanics of the harmonic Lagrange top is described by the most general confluent Heun equation (also known as the generalised spheroidal wave equation). We derive formulas for an infinite pentadiagonal symmetric matrix representing the Hamiltonian from which the spectrum is computed.

Highly Accurate Pseudospectral Approximations of the Prolate Spheroidal Wave Equation for Any Bandwidth Parameter and Zonal Wavenumber

The prolate spheroidal wave equation (PSWE) is transformed, using suitable mappings, into three different canonical forms which resemble the Jacobi, Laguerre and the Hermite differential equations. The eigenpairs of the PSWE are approximated with the corresponding classical orthogonal polynomial as a basis set. It is observed that for any zonal wavenumber m the Jacobi type pseudospectral methods are well suited for small bandwidth parameters c whereas the Hermite and Laguerre pseudospectral methods are appropriate for very large c values. Moreover, Jacobi pseudospectral methods work well for any parameter values such that $$m\ge c$$mźc. Our numerical results confirm that for any values of m, the Jacobi $$\left[ (\alpha ,\beta )=(\pm 1/2,m)\right] $$(ź,β)=(±1/2,m) and the Laguerre $$({\upgamma }=\pm 1/2)$$(ź=±1/2) pseudospectral methods formulated in this article for the numerical solution of the PSWE with small and very large bandwidth parameters, respectively, are highly efficient both from the accuracy and fastness point of view.

Calculation of the spheroidal functions of the first kind for complex values of the argument and parameters

Methods are proposed, first, for calculating (in a given domain of complex plane) the eigenvalues of the spheroidal wave equation with complex-valued parameters and, second, for calculating the values of the corresponding functions for complex values of the argument.

Integral relations for solutions of the confluent Heun equation

Firstly, we construct kernels for integral relations among solutions of the confluent Heun equation (CHE). Additional kernels are systematically generated by applying substitutions of variables. Secondly, we establish integral relations between known solutions of the CHE that are power series and solutions that are series of special functions. Thirdly, by using one of the integral relations as an integral transformation we obtain a new series solution of the ordinary spheroidal wave equation (a particular CHE). From this solution we construct new series solutions of the general CHE, and show that these are suitable for solving the radial part of the two-center problem in quantum mechanics.

Adaptive radial basis function and Hermite function pseudospectral methods for computing eigenvalues of the prolate spheroidal wave equation for very large bandwidth parameter

Asymptotic approximations show that the lowest modes of the prolate spheroidal wave equation are concentrated with an O(1/c) length scale where c is the “bandwidth” parameter of the prolate differential equation. Accurate computation of the ground state eigenvalue by the long-known Legendre–Galerkin method requires roughly 3.8c Legendre polynomials. Some studies have therefore applied a grid with 20,000 points in conjunction with high order finite differences to achieve c=107. Here, we show that by adaptively applying either Hermite functions or Gaussian radial basis functions (RBFs), it is never necessary to use more than eighty degrees of freedom to calculate the lowest dozen eigenvalues of each symmetry class. For small c, the eigenmodes are not confined to a small portion of the domain θ∈[−π/2,π/2] in latitude, but are global. We show that by periodizing the basis functions via imbricate series, it is possible to apply Hermite and RBFs even in the limit c→0. (The Legendre method is probably a little more efficient in this limit since the prolate functions tend to Legendre polynomials in this limit.) A “sideband truncation” restricts the discretization to a small block taken from the large Hermite–Galerkin matrix. We show that sideband truncation with blocks as small as 5×5 is a very efficient way to compute high order modes. In an appendix, we prove a rigorous convergence theorem for the periodized Hermite expansion.

A finite difference construction of the spheroidal wave functions

A finite difference construction of the spheroidal wave functions

Solving the spin-weighted spheroidal wave equation

In this paper we solve spin-weighted spheroidal wave equations through super-symmetric quantum mechanics with a different expression of the super-potential. We use the shape invariance property to compute the “excited" eigenvalues and eigenfunctions. The results are beneficial to researchers for understanding the properties of the spin-weighted spheroidal wave more deeply, especially its integrability.

Analytic solutions of the ground and excited states of the spin-weighted spheroidal equation in the case of s = 2

By using the super-symmetric quantum mechanics (SUSYQM) method, this paper obtains the analytical solutions for the spin-weighted spheroidal wave equation in the case of s = 2. Based on the derived W0 to W4 the general form for the n-th-order super-potential is summarized and is proved correct by mathematical induction. Hence the ground eigenvalue problem is completely solved. Particularly, the novel solutions of the excited state are investigated according to the shape-invariance property.

Solving the spheroidal wave equation with s=3/2 by super-symmetric quantum mechanic

In the paper, we use the method of super-symmetric quantum mechanics to study the spin-weighted spheroidal wave equation in the case of s=3/2. We first change the equation into Schrdinger's form, then calculate and derive the first four terms concerning E and super-potential W. we summarize the general formula of super-potential Wn and use mathematical induction to prove its correctness. In turn, this completely gives all the information about the ground eigenvalue and eigenfunction.

Low frequency asymptotics for the spin-weighted spheroidal equation in the case of s = 1/2

The spin-weighted spheroidal equation in the case of s = 1/2 is thoroughly studied by using the perturbation method from the supersymmetric quantum mechanics. The first-five terms of the superpotential in the series of parameter β are given. The general form for the n-th term of the superpotential is also obtained, which could also be derived from the previous terms Wk, k < n. From these results, it is easy to obtain the ground eigenfunction of the equation. Furthermore, the shape-invariance property in the series of parameter β is investigated and is proven to be kept. This nice property guarantees that the excited eigenfunctions in the series form can be obtained from the ground eigenfunction by using the method from the supersymmetric quantum mechanics. We show the perturbation method in supersymmetric quantum mechanics could completely solve the spin-weight spheroidal wave equations in the series form of the small parameter β.

Scalar Spheroidal Harmonics in Five Dimensional Kerr-(A)dS

We derive expressions for the general five-dimensional metric for Kerr-(A)dS black holes. The Klein-Gordon equation is explicitly separated and we show that the angular part of the wave equation leads to just one spheroidal wave equation, which is also that for charged five-dimensional Kerr-(A)dS black holes. We present results for the perturbative expansion of the angular eigenvalue in powers of the rotation parameters up to 6th order and compare numerically with the continued fraction method.

Spin-weighted spheroidal equation in the case of s = 1

We present a series of studies to solve the spin-weighted spheroidal wave equation by using the method of super-symmetric quantum mechanics. We first obtain the first four terms of super-potential of the spin-weighted spheroidal wave equation in the case of s = 1. These results may help summarize the general form for the n-th term of the super-potential, which is proved to be correct by means of induction. Then we compute the eigen-values and the eigenfunctions for the ground state. Finally, the shape-invariance property is proved and the eigen-values and eigenfunctions for excited states are obtained. All the results may be of significance for studying the electromagnetic radiation processes near rotating black holes and computing the radiation reaction in curved space-time.

Ground eigenvalue and eigenfunction of a spin-weighted spheroidal wave equation in low frequencies

Spin-weighted spheroidal wave functions play an important role in the study of the linear stability of rotating Kerr black holes and are studied by the perturbation method in supersymmetric quantum mechanics. Their analytic ground eigenvalues and eigenfunctions are obtained by means of a series in low frequency. The ground eigenvalue and eigenfunction for small complex frequencies are numerically determined.

Solving ground eigenvalue and eigenfunction of spheroidal wave equation at low frequency by supersymmetric quantum mechanics method

The spheroidal wave functions are found to have extensive applications in many branches of physics and mathematics. We use the perturbation method in supersymmetric quantum mechanics to obtain the analytic ground eigenvalue and the ground eigenfunction of the angular spheroidal wave equation at low frequency in a series form. Using this approach, the numerical determinations of the ground eigenvalue and the ground eigenfunction for small complex frequencies are also obtained.

Stokes phenomenon for the prolate spheroidal wave equation

The prolate spheroidal wave functions can be characterized as the eigenfunctions of a differential operator of order 2: ( t 2 − τ 2 ) φ ″ + 2 t φ ′ + σ 2 t 2 φ = μ φ . In this article we study the formal solutions of this equation in the neighborhood of the singularities (the regular ones ± τ, and the irregular one, at infinity) and perform some numerical experiments on the computation of Stokes matrices and monodromy, using formal/numerical algorithms we developed recently in the Maple package Desir. This leads to the following conjecture: the series appearing in the formal solutions at infinity, depending on the parameter μ, are in general divergent; they become convergent for some particular values of the parameter, corresponding exactly to the eigenvalues of the prolate operator. We give the proof of this result and its interpretation in terms of differential Galois groups.

A New Method to Solve the Spheroidal Wave Equations

The perturbation method in supersymmetric quantum mechanics is used to study the spheroidal wave functions' eigenvalue problem. The super-potential are solved in series of the parameter α, and the general form of all its terms is obtained. This means that the spheroidal problem is solved completely in the way for the ground eigen-value problem. The shape invariance property is proved retained for the super-potential and subsequently all the excited eigen-value problem could be solved. The results show that the spheroidal wave equations are integrable.

Grid method for computation of generalized spheroidal wave functions based on discrete variable representation

We present an efficient and accurate grid method for computations of eigenvalues and eigenfunctions of the generalized spheroidal wave equation. Different from previous studies, our method is based on the expansion of the spheroidal wave function by discrete-variable-representation basis functions constructed from the associated Legendre polynomials. The differential operator can be expressed analytically on the grid points, which are the zeros of the associated Legendre polynomials. The resultant potential matrix is simply diagonal and evaluated directly on the same grid. The corresponding differential equation is thus converted to an eigenvalue problem of a small matrix, whose eigenvalues and eigenvectors are converged very fast. The wave functions can then be evaluated accurately at any desired point from the expansion formula with the computed eigenvectors. Compared to previous methods, our method is direct and efficient for any parameter c , either small or large.

An approximation to solutions of linear ODE by cubic interpolation

We present a method of integration for non-autonomous non-homogeneous systems of linear ordinary differential equation (ODE), which is based in both, the cubic polynomial segmentary interpolation and the minimal square method. This method is valid for nonhomogeneous ordinary linear second order differential equations in the neighborhood of regular and singular regular points. We illustrate the method with the Mathieu and Bessel equations and two other equations that arise in the study of quantum systems with axial symmetry, which are versions of the spheroidal wave equation.

Calculating the branch points of the eigenvalues of the Coulomb spheroidal wave equation

A method for computing the eigenvalues λ mn (b, c) and the eigenfunctions of the Coulomb spheroidal wave equation is proposed in the case of complex parameters b and c. The solution is represented as a combination of power series expansions that are then matched at a single point. An extensive numerical analysis shows that certain b s and c s are second-order branch points for λ mn (b, c) with different indices n 1 and n 2, so that the eigenvalues at these points are double. Pade approximants, quadratic Hermite-Pade approximants, the finite element method, and the generalized Newton method are used to compute the branch points b s and c s and the double eigenvalues to high accuracy. A large number of these singular points are calculated.