Heun Functions Research Articles

Charged scalar fields in a Kerr–Sen black hole: exact solutions, Hawking radiation, and resonant frequencies * * H.S.V. is funded by the CNPq through the research Project (150640/2018-8). V.B.B. is partially supported by the CNPq through the research Project (305835/2016-5)

In this study, we consider charged massive scalar fields around a Kerr–Sen spacetime. The radial and angular parts of the covariant Klein–Gordon equation are solved in terms of the confluent Heun function. From the exact radial solution, we obtain the Hawking radiation spectrum and discuss its resonant frequencies. The massless case of the resonant frequencies is also examined.

Semi-exact Solutions of Konwent Potential**Supported by the project under Grant No. 20180677-SIP-IPN, COFAA-IPN, Mexico and partially by the CONACYT project under Grant No. 288856-CB-2016

In this work we study the quantum system with the symmetric Konwent potential and show how to find its exact solutions. We find that the solutions are given by the confluent Heun function. The eigenvalues have to be calculated numerically because series expansion method does not work due to the variable z ≥ 1. The properties of the wave functions depending on the potential parameter A are illustrated for given potential parameters V0 and a. The wave functions are shrunk towards the origin with the increasing |A|. In particular, the amplitude of wave function of the second excited state moves towards the origin when the positive parameter A decreases. We notice that the energy levels ϵi increase with the increasing potential parameter |A| ≥ 1, but the variation of the energy levels becomes complicated for |A| ∈ (0, 1), which possesses a double well. It is seen that the energy levels ϵi increase with |A| for the parameter interval A ∈ (−1, 0), while they decrease with |A| for the parameter interval A ∈ (0, 1).

Calculation of the Discrete Spectrum of some Two-Dimensional Schrödinger Equations with a Magnetic Field

One of us previously obtained and integrated the first examples of two-dimensional Schrodinger equations with a magnetic field belonging to the class of quasi–exactly solvable problems. It was shown that the wave functions are expressed in terms of degenerations of the Heun function: biconfluent and confluent Heun functions. Algebraic conditions were also found that determine the discrete spectrum and wave functions. Our goal here is to solve these algebraic equations numerically. In some cases, we can find an analytic approximation of the discrete spectrum.

Q-deformed superstatistics of the anharmonic oscillator for unrelativistic and relativistic (K–G equation) cases in noncommutative plane

In this article, we present the thermodynamical properties of the anharmonic oscillator for unrelativistic and relativistic K–G equations in the presence of an external magnetic field in noncommutative plane using q-deformed superstatistics. The explicit form of energy eigenvalues and the wave functions of the considered system are derived in terms of the bi-confluent Heun functions. Subsequently, by employing the effective Boltzmann factor, the thermodynamical properties of the anharmonic oscillator ensemble for both cases are obtained and, some graphs are plotted for the clarity of our results.

Saturation in heavy quarkonia spectra with energy-dependent confining potential inN-dimensional space

We study the energy spectra of Schrödinger Hamiltonian in N-dimensional space with Coulomb plus inverse quadratic term [Formula: see text] and energy-dependent confining potential of the form [Formula: see text]. It is found that the energy spectra exhibit saturation effect, where these spectra are suppressed to finite values at large quantum numbers n and/or [Formula: see text] at any dimensional space N. As a typical application, we obtained a compact mass formula for the spectra of heavy quarkonia ([Formula: see text], [Formula: see text]). It is found that the model is capable of reproducing the mass spectra of these systems with observed saturation governed by a factor [Formula: see text]. The comparison of this work predictions with available experimental data as well as other relevant theoretical works is also presented. The eigenfunctions are expressed in terms of the bi-confluent Heun functions.

New indefinite integrals of confluent Heun functions

ABSTRACTWe obtain some new formulae to compute the first derivative of confluent and biconfluent Heun functions under the minimal assumption of fixing only one parameter. These results together with the Lagrangian formulation of a general homogeneous linear ordinary differential equation allow to construct several new indefinite integrals for the confluent, biconfluent, doubly confluent, and triconfluent Heun functions.

Exact Solutions of the Razavy Cosine Type Potential

We solve the quantum system with the symmetric Razavy cosine type potential and find that its exact solutions are given by the confluent Heun function. The eigenvalues are calculated numerically. The properties of the wave functions, which depend on the potential parameter a, are illustrated for a given potential parameter ξ. It is shown that the wave functions are shrunk to the origin when the potential parameter a increases. We note that the energy levels ϵi (i∈[1,3]) decrease with the increasing potential parameter a but the energy levels ϵi (i∈[4,7]) first increase and then decrease with the increasing a.

Erratum to: Analytical solution of Bohr Hamiltonian and extended form of sextic potential using bi-confluent Heun functions

Erratum to: Analytical solution of Bohr Hamiltonian and extended form of sextic potential using bi-confluent Heun functions

Propagation of electromagnetic waves through non-uniform dielectric layers

The propagation of TE- and TM-polarized electromagnetic (EM) waves through a dielectric layer with spatial variation of the refractive index is reported in this paper. The relative permittivity of the layer is assumed to be graded along the thickness direction, and its spatial variation is described by a quartic polynomial. The corresponding mode equations are Helmholtz equations with variable coefficients that can be transformed to a triconfluent Heun equation, a local exact solution of which is expressed in terms of triconfluent Heun functions. The solutions cover many particular cases owing to a variability of four optional parameters (coefficients) of the quartic polynomial. The general local solution for a TE-polarized EM wave is employed for the calculation of transmission properties of a periodic one-dimensional photonic crystal using the Floquet theory.

Analytical solutions in a cosmic string Born–Infeld-dilaton black hole geometry: quasinormal modes and quantization

Chargeless massive scalar fields are studied in the spacetime of Born-Infeld dilaton black holes (BIDBHs). We first separate the massive covariant Klein-Gordon equation into radial and angular parts and obtain the exact solution of the radial equation in terms of the confluent Heun functions. Using the obtained radial solution, we show how one gets the exact quasinormal modes (QNMs) for some particular cases. We also solve the Klein-Gordon equation in the spacetime of BIDBH with a cosmic string in which we point out the effect of the conical deficit on the BIDBH. The analytical solutions of the radial and angular parts are obtained in terms of the confluent Heun functions. Finally, we study the quantization of the BIDBH. While doing this, we also discuss the Hawking radiation in terms of the effective temperature.

A Riemann-Hilbert Approach to the Heun Equation

We describe the close connection between the linear system for the sixth Painlev\'e equation and the general Heun equation, formulate the Riemann-Hilbert problem for the Heun functions and show how, in the case of reducible monodromy, the Riemann-Hilbert formalism can be used to construct explicit polynomial solutions of the Heun equation.

Semiexact Solutions of the Razavy Potential

In this work, we study the quantum system with the symmetric Razavy potential and show how to find its exact solutions. We find that the solutions are given by the confluent Heun functions. The eigenvalues have to be calculated numerically. The properties of the wave functions depending on m are illustrated graphically for a given potential parameter ξ. We find that the even and odd wave functions with definite parity are changed to odd and even wave functions when the potential parameter m increases. This arises from the fact that the parity, which is a defined symmetry for very small m, is completely violated for large m. We also notice that the energy levels ϵi decrease with the increasing potential parameter m.

Heun Functions and Some of Their Applications in Physics

Most of the theoretical physics known today is described by using a small number of differential equations. For linear systems, different forms of the hypergeometric or the confluent hypergeometric equations often suffice to describe the system studied. These equations have power series solutions with simple relations between consecutive coefficients and/or can be represented in terms of simple integral transforms. If the problem is nonlinear, one often uses one form of the Painlevé equations. There are important examples, however, where one has to use higher order equations. Heun equation is one of these examples, which recently is often encountered in problems in general relativity and astrophysics. Its special and confluent forms take names as Mathieu, Lamé, and Coulomb spheroidal equations. For these equations whenever a power series solution is written, instead of a two-way recursion relation between the coefficients in the series, we find one between three or four different ones. An integral transform solution using simpler functions also is not obtainable. The use of this equation in physics and mathematical literature exploded in the later years, more than doubling the number of papers with these solutions in the last decade, compared to time period since this equation was introduced in 1889 up to 2008. We use SCI data to conclude this statement, which is not precise, but in the correct ballpark. Here this equation will be introduced and examples for its use, especially in general relativity literature, will be given.

New indefinite integrals of Heun functions

ABSTRACTWe present a conspicuous number of indefinite integrals involving Heun functions and their products obtained by means of the Lagrangian formulation of a general homogeneous linear ordinary differential equation. As a by-product we also derive new indefinite integrals involving the Gauss hypergeometric function and products of hypergeometric functions with elliptic functions of the first kind. All integrals we obtained cannot be computed using Maple and Mathematica.

Expansions of the Solutions of the General Heun Equation Governed by Two-Term Recurrence Relations for Coefficients

We examine the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions. We present several expansions using functions, the forms of which differ from those applied before. In general, the coefficients of the expansions obey three-term recurrence relations. However, there exist certain choices of the parameters for which the recurrence relations become two-term. The coefficients of the expansions are then explicitly expressed in terms of the gamma functions. Discussing the termination of the presented series, we show that the finite-sum solutions of the general Heun equation in terms of generally irreducible hypergeometric functions have a representation through a single generalized hypergeometric function. Consequently, the power-series expansion of the Heun function for any such case is governed by a two-term recurrence relation.

Elliptic basis for the Zernike system: Heun function solutions

The differential equation that defines the Zernike system, originally proposed to classify wavefront aberrations of the wavefields in the disk of a circular pupil, had been shown to separate in three distinct coordinate systems obtained from polar coordinates on a half-sphere. Here we find and examine the separation in the generic elliptic coordinate system on the half-sphere and its projected disk, where the solutions, separated in Jacobi coordinates, contain Heun polynomials.

Analytical results for periodically-driven two-level models in relation to Heun functions

We introduce three different types of periodically-driven multiparametric two-level models whose analytical solutions are given in terms of Heun functions. These results are applied to obtain exact analytical results for certain types of periodic potentials and asymmetric double-well potentials. In particular, it is shown that under special parameter conditions, an experimentally realised periodic potential supports the exact in-gap solutions. In the asymmetric double-well potentials, some exact results of the bound-state wave functions and associated energies are found in explicit form.

Constructions of the Soluble Potentials for the Nonrelativistic Quantum System by Means of the Heun Functions

The Schrödinger equation ψ′′(x)+κ2ψ(x)=0 where κ2=k2-V(x) is rewritten as a more popular form of a second order differential equation by taking a similarity transformation ψ(z)=ϕ(z)u(z) with z=z(x). The Schrödinger invariant IS(x) can be calculated directly by the Schwarzian derivative z,x and the invariant I(z) of the differential equation uzz+f(z)uz+g(z)u=0. We find an important relation for a moving particle as ∇2=-IS(x) and thus explain the reason why the Schrödinger invariant IS(x) keeps constant. As an illustration, we take the typical Heun’s differential equation as an object to construct a class of soluble potentials and generalize the previous results by taking different transformation ρ=z′(x) as before. We get a more general solution z(x) through integrating (z′)2=α1z2+β1z+γ1 directly and it includes all possibilities for those parameters. Some particular cases are discussed in detail. The results are also compared with those obtained by Bose, Lemieux, Batic, Ishkhanyan, and their coworkers. It should be recognized that a subtle and different choice of the transformation z(x) also related to ρ will lead to difficult connections to the results obtained from other different approaches.

Investigation of the Dirac Equation by Using the Conformable Fractional Derivative

In this paper,the Dirac equation is constructed using the conformable fractional derivative so that in its limit for the fractional parameter, the normal version is recovered. Then, the Cornell potential is considered as the interaction of the system. In this case, the wave function and the energy eigenvalue equation are derived with the aim of the bi-confluent Heun functions. use of the conformable fractional derivative is proven to lead to a branching treatment for the energy of the system. Such a treatment is obvious for small values of the fractional parameter, and a united value as the fractional parameter approaches unity.

One-dimensional propagation of longitudinal elastic waves through functionally graded materials

The one-dimensional propagation of longitudinal elastic waves along the thickness of a plate made of functionally graded materials excited by a harmonic force is reported in this article. The material properties of the plate are assumed to be graded along the thickness direction according to a trigonometric law distribution. This distribution smoothly connects the material properties of the upper and lower homogeneous materials that bounds the plate. The corresponding propagation equation is Ince-type equation that can be transformed to Heun’s equation a local exact solution of which is expressed in terms of local Heun functions. The general nature of these functions is demonstrated based on four degenerate cases of Heun’s equation. The transfer matrix method is used to study the elastic waves propagating in the inhomogeneous domain. The calculation of the transfer matrices requires the evaluation of the general solution in the interval containing two regular singular points. For this purpose, the modified Heun function is introduced. Based on the transfer matrices, the influence of both the asymmetry of the unit cell and various constituent materials on the transmission coefficient spectrum is studied. The transmission coefficient is also calculated for the locally periodic structures with the help of the Chebyshev polynomials.