Radial Eigenfunctions Research Articles

On the Five-Dimensional Non-Extremal Reissner–Nordström Black Hole: Retractions and Scalar Quasibound States

In this paper, we examine the role played by topology, and some specific boundary conditions as well, on the physics of a higher-dimensional black hole. We analyze the line element of a five-dimensional non-extremal Reissner–Nordström black hole to obtain a new family of subspaces that are types of strong retractions and deformations, and then we extend these results to higher dimensions in order to deduce the relationship between various types of transformations. We also study the scalar field perturbations in the background under consideration and obtain an analytical expression for the quasibound state frequencies by using the Vieira–Bezerra–Kokkotas approach, which uses the polynomial conditions of the general Heun functions, and then we discuss the stability of the system and present the radial eigenfunctions. Our main goal is to discuss the physical meaning of these mathematical applications in such higher-dimensional effective metric.

The Partition of Unity Finite Element Method for the Schrödinger Equation

Abstract A Schrödinger equation for the system’s wavefunctions in a parallelepiped unit cell subject to Bloch-periodic boundary conditions must be solved repeatedly in quantum mechanical computations to derive the materials’ properties. Recent studies have demonstrated how enriched finite element type Galerkin methods can substantially lower the number of degrees of freedom necessary to produce accurate solutions with respect to the standard plane-waves method. In particular, the flat-top partition of unity finite element method enriched with the radial eigenfunctions of the one-dimensional Schrödinger equation offers a very effective way of solving the three-dimensional Schrödinger eigenvalue problem. We investigate the theoretical properties of this approximation method, its well-posedness and stability, we prove its convergence and derive suitable bound for the ℎ- and 𝑝-refinement in the L 2 L^{2} and energy norm for both the eigenvalues and the eigenfunctions. Finally, we confirm these theoretical results by applying this method to the eigenvalue problem of the one-electron Schrödinger equation with the harmonic potential, for which the exact solution is known.

Control of chaos in Darcy–Bénard axisymmetric convection in a cylindrical enclosure using a uniform vertical cross-flow

The linear and weakly nonlinear stability analyses of Darcy–Bénard convection of a Newtonian fluid experiencing a uniform vertical cross-flow is investigated in the paper for various aspect ratios. Making use of the Maclaurin series representation, an expression for axial eigenfunctions is obtained with the radial eigenfunction being a Bessel function of first kind. These eigenfunctions are influenced by the Peclet number, Pe, the non-dimensional number that signifies the rate of vertical cross-flow. The modified-Vadasz-Lorenz model obtained in this paper has newly defined non-dimensional parameters that capture the influence of vertical cross-flow. From the linear stability analysis, it is found that the effect of introducing vertical cross-flow is to stabilize the system. Using a weakly nonlinear stability analysis, the closed-form expression of the Hopf-Rayleigh number as a function of Pe is obtained. Furthermore, the behavior of the modified-Vadasz–Lorenz model is analyzed using the largest Lyapunov exponent and the bifurcation diagram. This gives information about the intensity of chaos and occurrence of the periodic motion. We observe that the influence of vertical cross-flow is to increase the value of the Hopf–Rayleigh number and thereby to delay the onset of chaos. Furthermore, the appearance of the first periodic point is preponed due to the vertical cross-flow. As the rate of vertical cross-flow increases, the intensity of chaos decreases, thereby indicating that the effect of introducing vertical cross-flow is to suppress chaos.

Quasibound states of analytic black-hole configurations in three and four dimensions

In this work we analyze the sound perturbation of Unruh's acoustic effective geometry in both (2+1) and (3+1) spacetime dimensions and present an exact analytical expression for the quasibound states of these idealized black-hole configurations by using a new approach recently developed, which uses the polynomial conditions of the hypergeometric functions. Our main goal is to discuss the effects of having an event horizon in such effective metrics. We also discuss the stability of the systems and present the radial eigenfunctions related to these quasibound state frequencies. These metrics assume just the form it has for a Schwarzschild black hole near the event horizon, and therefore may, in principle, shed some light into the underlying classical and quantum physics of astrophysical black holes through analog acoustic probes.

Nonradiality of second fractional eigenfunctions of thin annuli

In the present paper, we study properties of the second Dirichlet eigenvalue of the fractional Laplacian of annuli-like domains and the corresponding eigenfunctions. In the first part, we consider an annulus with inner radius $ R $ and outer radius $ R+1 $. We show that for $ R $ sufficiently large any corresponding second eigenfunction of this annulus is nonradial. In the second part, we investigate the second eigenvalue in domains of the form $ B_1(0)\setminus \overline{B_{\tau}(a)} $, where $ a $ is in the unitary ball and $ 0<\tau<1-|a| $. We show that this value is maximized for $ a = 0 $, if the set $ B_1(0)\setminus \overline{B_{\tau}(0)} $ has no radial second eigenfunction. We emphasize that the first part of our paper implies that this assumption is indeed nonempty.

Emergence of radial orientation selectivity: Effect of cell density changes and eccentricity in a layered network.

We establish a simple mechanism by which radially oriented simple cells can emerge in the primary visual cortex. In 1986, R. Linsker. proposed a means by which radially symmetric, spatial opponent cells can evolve, driven entirely by noise, from structure in the initial synaptic connectivity distribution. We provide an analytical derivation of Linsker's results, and further show that radial eigenfunctions can be expressed as a weighted sum of degenerate Cartesian eigenfunctions, and vice-versa. These results are extended to allow for radially dependent cell density, from which we show that, despite a circularly symmetric synaptic connectivity distribution, radially biased orientation selectivity emerges in the third layer when cell density in the first layer, or equivalently, synaptic radius, changes with eccentricity; i.e., distance to the center of the lamina. This provides a potential mechanism for the emergence of radial orientation in the primary visual cortex before eye opening and the onset of structured visual input after birth.

An isoperimetric inequality for the perturbed Robin bi-Laplacian in a planar exterior domain

We introduce the perturbed two-dimensional Robin bi-Laplacian in the exterior of a planar bounded simply-connected C2-smooth open set. The considered lower-order perturbation corresponds to tension. We prove that the essential spectrum of this operator coincides with the positive semi-axis and that the negative discrete spectrum is non-empty if the boundary parameter is negative. As the main result, we obtain an isoperimetric inequality for the lowest eigenvalue of such a perturbed Robin bi-Laplacian with a negative boundary parameter in the exterior of a bounded convex planar set under the constraint on the maximum of the curvature of the boundary with the maximizer being the exterior of the disk. The isoperimetric inequality is proved under the assumption that to the lowest eigenvalue for the exterior of the disk corresponds a radial eigenfunction. We provide a sufficient condition in terms of the tension parameter and the radius for this property to hold.

Instability of the charged massive scalar field on the Kerr–Newman black hole spacetime

We investigate the quasibound states of charged massive scalar fields in the Kerr–Newman black hole spacetime by using a new approach recently developed, which uses the polynomial conditions of the Heun functions. We calculate the resonant frequencies related to the spectrum of quasibound states, as well as its corresponding angular and radial wave eigenfunctions. We also analyze the instability of the system. These results are particularized to the cases of Schwarzschild and Kerr black holes. Additionally, we compare our analytical results with the numerical ones known in the literature. Finally, we apply the obtained results to compute the characteristic times of growth and decay of bosonic particles around a supermassive black hole situated at the center of the M87 galaxy.

An inequality of Bessel functions and applications to transcritical bifurcation problems of nonlinear elliptic equations

We prove a weighted integral inequality of the Bessel function of zero order, which implies the positivity of the integral of the cube of nodal radial eigenfunctions of the Laplacian operator in the planar disk. This solves an open problem in Bonheure et al. (2016) and Corsato et al. (2020). The positivity plays a crucial role in transcritical bifurcation problems of nodal radial solutions of nonlinear elliptic equations.

II. The effect of axisymmetric and spatially varying equilibria and flow on MHD wave modes: cylindrical geometry

ABSTRACT Magnetohydrodynamic (MHD) waves are routinely observed in the solar atmosphere. These waves are important in the context of solar physics as it is widely believed they can contribute to the energy budget of the solar atmosphere and are a prime candidate to contribute towards coronal heating. Realistic models of these waves are required representing observed configurations such that plasma properties can be determined more accurately, since they cannot be measured directly. This work utilizes a previously developed numerical technique to find permittable eigenvalues under different non-uniform equilibrium conditions in a Cartesian magnetic slab geometry. Here, we investigate the properties of magnetoacoustic waves under non-uniform equilibria in a cylindrical geometry. Previously obtained analytical results are retrieved to emphasize the power and applicability of this numerical technique. Further case studies investigate the effect that a radially non-uniform plasma density and non-uniform plasma flow, modelled as a series of Gaussian profiles, have on the properties of different MHD waves. For all cases the dispersion diagrams are obtained and spatial eigenfunctions calculated which display the effects of the equilibrium inhomogeneity. It is shown that as the equilibrium non-uniformity is increased, the radial spatial eigenfunctions are affected and extra nodes introduced, similar to the previous investigation of a magnetic slab. Furthermore, azimuthal perturbations are increased with increasing inhomogeneity introducing vortical motions inside the waveguide. Finally, 2D and 3D representations of the velocity fields are shown which may be useful for observers for wave mode identification under realistic magnetic waveguides with ever increasing instrument resolution.

No-short scalar hair theorem for spinning acoustic black holes in a photon-fluid model

It has recently been revealed that spinning black holes of the photon-fluid model can support acoustic `clouds', stationary density fluctuations whose spatially regular radial eigenfunctions are determined by the $(2+1)$-dimensional Klein-Gordon equation of an effective massive scalar field. Motivated by this intriguing observation, we use {\it analytical} techniques in order to prove a no-short hair theorem for the composed acoustic-black-hole-scalar-clouds configurations. In particular, it is proved that the effective lengths of the stationary bound-state co-rotating acoustic scalar clouds are bounded from below by the series of inequalities $r_{\text{hair}}>{{1+\sqrt{5}}\over{2}}\cdot r_{\text{H}}>r_{\text{null}}$, where $r_{\text{H}}$ and $r_{\text{null}}$ are respectively the horizon radius of the supporting black hole and the radius of the co-rotating null circular geodesic that characterizes the acoustic spinning black-hole spacetime.

ENERGY SPECTRUM AND SOME USEFUL EXPECTATION VALUES OF THE TIETZ-HULTHÉN POTENTIAL

In this paper, concept of supersymmetric quantum mechanics has been employed to derive expression for bound state energy eigenvalues of the Tietz-Hulthén potential, the corresponding equation for normalized radial eigenfunctions were deduced by ansatz solution technique. In dealing with the centrifugal term of the effective potential of the Schrödinger equation, a Pekeris-like approximation recipe is considered. By means of the expression for bound state energy eigenvalues and radial eigenfunctions, equations for expectation values of inverse separation-squared and kinetic energy of the Tietz-Hulthén potential were obtained from the Hellmann-Feynman theorem. Numerical values of bound state energy eigenvalues and expectation values of inverse separation-squared and kinetic energy the Tietz-Hulthén potential were computed at arbitrary principal and angular momentum quantum numbers. Results obtained for computed energy eigenvalues of Tietz-Hulthén potential corresponding to Z = 0 and V0 = 0 are in excellent agreement with available literature data for Tietz and Hulthén potentials respectively. Studies have also revealed that increase in parameter Z results in monotonic increase in the mean kinetic energy of the system. The results obtained in this work may find suitable applications in areas of physics such as: atomic physics, chemical physics, nuclear physics and solid state physics

Harmonic radial vector fields on harmonic spaces

We characterize harmonic spaces in terms of the dimensions of various spaces of radial eigen-spaces of the Laplacian Δ0 on functions and the Laplacian Δ1 on 1-forms. We examine the nature of the singularity as the geodesic distance r tends to zero of radial eigen-functions and 1-forms. Via duality, our results give rise to corresponding results for radial vector fields.

J—state solutions and thermodynamic properties of the Tietz oscillator

In this work, we have solved the radial part of the Schrödinger equation with Tietz potential to obtain explicit expressions for bound state ro-vibrational energies and radial eigenfunctions. The proper quantization rule and ansatz solution technique were used to arrive at the solutions. In modeling the pseudo-spin–orbit term of the effective potential, the Pekeris-like and the Greene-Aldrich approximation recipes were applied. Using our equation for eigen energies, we have deduced expression for bound state energy eigenvalues of Deng-Fan oscillator. The result obtained agrees with available literature data for this potential. Also, for arbitrary values of rotational and vibrational quantum numbers, we have calculated bound state energies for the Tietz oscillator. Our computed results are in excellent agreement with those in the literature. Furthermore, the result showed that unlike Greene-Aldrich approximation, energies computed based on Pekeris-like approximation are better and almost indistinguishable from numerically obtained energies of the Tietz oscillator in the literature. With the help of our formula for ro-vibrational energy, analytical expressions for some important thermodynamic relations were also derived for the Tietz oscillator. The derived thermal functions which include ro-vibrational: partition function, free energy, mean energy, entropy and specific heat capacity were subsequently applied to the spectroscopic data of KI diatomic molecule. Studies of the thermal functions indicated that the partition function decreases monotonically as the temperature is raised and increases linearly for increase in the upper bound vibrational quantum number. On the other hand, increase in either temperature or upper bound vibrational quantum number amounts to monotonic rise in the entropy of the KI molecules

Algebraic derivation of Kramers–Pasternack relations based on the Schrödinger factorization method

The Kramers–Pasternack relations are used to compute the moments of r (both positive and negative) for all radial energy eigenfunctions of hydrogenic atoms. They consist of two algebraic recurrence relations, one for positive powers and one for negative. Most derivations employ the Feynman–Hellmann theorem or a brute-force integration to determine the second inverse moment, which is needed to complete the recurrence relations for negative moments. In this work, we show both how to derive the recurrence relations algebraically and how to determine the second inverse moment algebraically, which removes the pedagogical confusion associated with differentiating the Hamiltonian with respect to the angular momentum quantum number l, which is restricted to be an integer, in order to find the inverse second moment.

Approximate analytical solutions and mean energies of stationary Schrödinger equation for general molecular potential

The Schrödinger equation is solved with general molecular potential via the improved quantization rule. Expression for bound state energy eigenvalues, radial eigenfunctions, mean kinetic energy, and potential energy are obtained in compact form. In modeling the centrifugal term of the effective potential, a Pekeris-like approximation scheme is applied. Also, we use the Hellmann–Feynman theorem to derive the relation for expectation values. Bound state energy eigenvalues, wave functions and meanenergies of Woods–Saxon potential, Morse potential, Möbius squared and Tietz–Hua oscillators are deduced from the general molecular potential. In addition, we use our equations to compute the bound state energy eigenvalues and expectation values for four diatomic molecules viz. H2, CO, HF, and O2. Results obtained are in perfect agreement with the data available from the literature for the potentials and molecules. Studies also show that as the vibrational quantum number increases, the mean kinetic energy for the system in a Tietz–Hua potential increases slowly to a threshold value and then decreases. But in a Morse potential, the mean kinetic energy increases linearly with vibrational quantum number increasing.

Path integral approach to theD-dimensional quantum mechanics of the nonrelativistic Snyder–de Sitter model

In the context of Snyder–de Sitter (SdS) algebra, we formulate the D-dimensional momentum space path integral transition amplitude for harmonic oscillator and free particle. The exact energy spectrum and the corresponding normalized radial momentum space eigenfunctions are obtained through the different quantum corrections rule.

Optimal estimates for hyperbolic Poisson integrals of functions in [formula omitted] with [formula omitted] and radial eigenfunctions of the hyperbolic Laplacian

Assume that p∈(1,∞] and u=Ph[ϕ], where ϕ∈Lp(Sn−1,Rn). Then for any x∈Bn, we obtain the sharp inequalities |u(x)|≤Cq1q(x)(1−|x|2)n−1p‖ϕ‖Lpand|u(x)|≤Cq1q(1−|x|2)n−1p‖ϕ‖Lp for some function Cq(x) and constant Cq in terms of Gauss hypergeometric and Gamma functions, where q is the conjugate of p. This result generalizes and extends some known results from harmonic mapping theory (Kalaj and Marković (2012, Theorems 1.1 and 1.2) and Axler et al. (1992, Proposition 6.16)). The proofs are mainly based on certain characterizations of the radial eigenfunctions of the hyperbolic Laplacian Δh, which are of independent interest.

Rossby modes in slowly rotating stars: depth dependence in distorted polytropes with uniform rotation

Context.Large-scale Rossby waves have recently been discovered based on measurements of horizontal surface and near-surface solar flows.Aims.We are interested in understanding why it is only equatorial modes that are observed and in modelling the radial structure of the observed modes. To this aim, we have characterised the radial eigenfunctions ofrmodes for slowly rotating polytropes in uniform rotation.Methods.We followed Provost et al. (1981, A&A, 94, 126) and considered a linear perturbation theory to describe quasi-toroidal stellar adiabatic oscillations in the inviscid case. We used perturbation theory to write the solutions to the fourth order in the rotational frequency of the star. We numerically solved the eigenvalue problem, concentrating on the type of behaviour exhibited where the stratification is nearly adiabatic.Results.We find that for free-surface boundary conditions on a spheroid of non-vanishing surface density,rmodes can only exist forℓ = mspherical harmonics in the inviscid case and we compute their depth dependence and frequencies to leading order. For quasi-adiabatic stratification, the sectoral modes with no radial nodes are the only modes which are almost toroidal and the depth dependence of the corresponding horizontal motion scales asrm. For allrmodes, except the zero radial order sectoral ones, non-adiabatic stratification plays a crucial role in the radial force balance.Conclusions.The lack of quasi-toroidal solutions when stratification is close to neutral, except for the sectoral modes without nodes in radius, follows from the need for both horizontal and radial force balance. In the absence of super- or sub-adiabatic stratification and viscosity, both the horizontal and radial parts of the force balance independently determine the pressure perturbation. The only quasi-toroidal cases in which these constraints on the pressure perturbation are consistent are the special cases whereℓ = mand the horizontal displacement scales withrm.

Exploring the latitude and depth dependence of solar Rossby waves using ring-diagram analysis

Context. Global-scale equatorial Rossby waves have recently been unambiguously identified on the Sun. Like solar acoustic modes, Rossby waves are probes of the solar interior. Aims. We study the latitude and depth dependence of the Rossby wave eigenfunctions. Methods. By applying helioseismic ring-diagram analysis and granulation tracking to observations by HMI aboard SDO, we computed maps of the radial vorticity of flows in the upper solar convection zone (down to depths of more than 16 Mm). The horizontal sampling of the ring-diagram maps is approximately 90 Mm (∼7.5°) and the temporal sampling is roughly 27 hr. We used a Fourier transform in longitude to separate the different azimuthal orders m in the range 3 ≤ m ≤ 15. At each m we obtained the phase and amplitude of the Rossby waves as functions of depth using the helioseismic data. At each m we also measured the latitude dependence of the eigenfunctions by calculating the covariance between the equator and other latitudes. Results. We conducted a study of the horizontal and radial dependences of the radial vorticity eigenfunctions. The horizontal eigenfunctions are complex. As observed previously, the real part peaks at the equator and switches sign near ±30°, thus the eigenfunctions show significant non-sectoral contributions. The imaginary part is smaller than the real part. The phase of the radial eigenfunctions varies by only ±5° over the top 15 Mm. The amplitude of the radial eigenfunctions decreases by about 10% from the surface down to 8 Mm (the region in which ring-diagram analysis is most reliable, as seen by comparing with the rotation rate measured by global-mode seismology). Conclusions. The radial dependence of the radial vorticity eigenfunctions deduced from ring-diagram analysis is consistent with a power law down to 8 Mm and is unreliable at larger depths. However, the observations provide only weak constraints on the power-law exponents. For the real part, the latitude dependence of the eigenfunctions is consistent with previous work (using granulation tracking). The imaginary part is smaller than the real part but significantly nonzero.