WKB Approximation Research Articles

Quasinormal modes and shadow of Schwarzschild Black Holes embedded in a Dehnen Type Dark Matter Halo exhibiting string cloud

Abstract In this paper, we investigate a static spherically symmetric black hole (BH) embedded in a Dehnen-(1,4,0) type dark matter (DM) halo in the presence of a cloud string. We examine and presente data on how the core density of the DM halo parameter and the cloud string parameter affect BH attributes such as quasinormal modes (QNMs) and shadow cast. To do this, we first look into the effective potential of perturbation equations for three types of perturbation fields with different spins: massless scalar field, electromagnetic field, and gravitational field. Then, using the 6th order WKB approximation, we examine quasinormal modes of the BH disturbed by the three fields and derive quasinormal frequencies. The changes of QNM versus the core density parameter and the cloud string parameter for three disturbances are explored. We also investigate how the core density and the cloud string parameters affect the photon sphere and shadow radius. Interestingly, the study shows that the influence of Dehnen type DM and cloud string increases both photon spheres and shadow radius. Finally, we employ observational data from Sgr $A{^\star}$ and $M87{^\star}$ to set limitations on the BH parameters.

Out-of-time-ordered-correlators for the pure inverted quartic oscillator: classical chaos meets quantum stability

Out-of-time-ordered-correlators (OTOCs) have been suggested as a means to diagnose chaotic behavior in quantum mechanical systems. Recently, it was found that OTOCs display exponential growth for the inverted quantum harmonic oscillator, mirroring the fact that this system is classically and quantum mechanically unstable. In this work, I study OTOCs for the inverted anharmonic (pure quartic) oscillator in quantum mechanics, finding only oscillatory behavior despite the classically unstable nature of the system. For higher temperature, OTOCs seem to exhibit saturation consistent with a value of –2⟨x2⟩T ⟨p2⟩T at late times. I provide analytic evidence from the spectral zeta-function and the WKB method as well as direct numerical solutions of the Schrödinger equation that the inverted quartic oscillator possesses a real and positive energy eigenspectrum, and normalizable wave-functions.

Numerical investigation of simply supported periodic rectangular plates with acoustic black hole wedge

In this paper, we propose the periodic rectangle plate with one-dimensional acoustic black hole (ABH) wedge under the simply supported condition. The vibration performances have been investigated by the semi-analytic method based on the levy solution method combined with WKB approximation, which is also validated by finite element method (FEM) simulation. For the rectangle ABH wedge, we derive the flexural wave dispersive relationship in plate, which also revealed the cut-on frequency. It can also be found that with the increase of periodic number, the vibration of plate structure can be reduced not only in the high frequency range where traditional ABH effect takes effect, but in low and middle frequency range the periodic ABH wedge can also show its advantage. Meanwhile, the damping coupling performance between the periodic ABH wedge and attached damping layers has also been investigated to enhance the vibration reduction. This novel kind ABH structure shows its potential in engineering applications.

Approximate resolutions of the Schrodinger theory applying the WKB approximation for certain diatomic molecular interactions.

The Analyzing of energetic bond spectra of diatomic compounds is crucial to understanding theirqualities because it allows one to evaluate their attributes. Diatoms compounds' spectral properties andbound energies are presented in this study. These energies are found by solving the Schrodinger equationwhile making consideration of the employing of the Kratzer Feus potential. This study focuses on the calculation of bound states for diatomic molecules using the WKBapproximation. The final energy spectrum equation is utilized to compute the bound states of specificdiatomic molecules for varying quantum numbers n and l through the utilization of the Mathematicasoftware. The method produced the desired and anticipated results, as shown by a comparison of theeigenvalue results with earlier studies.

Quasinormal modes, thermodynamics and shadow of black holes in Hu–Sawicki f(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{f(R)}$$\\end{document} gravity theory

We derive novel black hole solutions in a modified gravity theory, namely the Hu–Sawicki model of f(R) gravity. After obtaining the black hole solution, we study the horizon radius of the black hole from the metric and then analyse the dependence of the model parameters on the horizon. We then use the 6th order WKB method to study the quasinormal modes of oscillations (QNMs) of the black hole perturbed by a scalar field. The dependence of the amplitude and damping part of the QNMs are analysed with respect to variations in model parameters and the error associated with the QNMs are also computed. After that we study some thermodynamic properties associated with the black hole such as its thermodynamic temperature as well as greybody factors. It is found that the black hole has the possibility of showcasing negative temperatures and is thermodynamically unstable for feasible values of model parameters. Then we analyse the geodesics and derive the photon sphere radius as well as the shadow radius of the black hole. The photon radius is independent of the model parameters while shadow radius showed fair amount of dependence on the model parameters. We tried to constrain the parameters with the help of Keck and VLTI observational data and obtained some bounds on m and c2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_{2}$$\\end{document} parameters.

Field intensity dependence of the dissociative multiple ionization of argon dimers in strong femtosecond laser fields

The dissociative ionization of Ar dimers is investigated in femtosecond laser fields with intensities from 260 to 1020 TW/cm2. The three-dimensional momentum and kinetic-energy release of fragmental ions generated from the channels Ar22+→Ar++Ar+, Ar23+→Ar2++Ar+, and Ar24+→Ar2++Ar2+ were measured with a cold-target recoil-ion momentum spectrometer. It is shown that the laser intensity significantly modulates the kinetic energies and angular distributions of fragmental ions from dissociative double ionization. Laser-induced charge-transfer following one-site double ionization contributes relatively more to the dissociative double ionization at lower laser intensity. The calculation results of a one-dimensional model based on the WKB approximation suggest that the charge transfer is suppressed at higher laser intensity due to the core polarization effect. In addition, double, triple, and quadruple dissociative ionizations of Ar dimers are accompanied by frustrated-tunneling ionization that increases with the laser intensity.

Quasinormal modes of the Mannheim–Kazanas black holes

Abstract A spherically symmetric black hole solution in the conformal Weyl gravity was found by Mannheim and Kazanas in 1988. While the quasinormal modes (QNMs) of these black holes have been considered in a few works, here we complement these studies by considering in detail the regime of vanishing cosmological constant and negative values of the Weyl parameter for which we find quasinormal frequencies for scalar, Dirac and electromagnetic perturbations with the time-domain integration and WKB methods. In particular, we derive the compact and remarkably accurate analytic formula for the frequencies in the form of expansion in terms of the inverse multipole number. Comparison with the time-domain integration shows that the 6th order WKB method with the Padé approximants is quite accurate, unless the black hole is in the near extreme state.

Influence of Bed Variations on Linear Wave Propagation beyond the Mild Slope Condition

As water waves travel from deep to shallow waters, they experience increased nonlinearity and decreased dispersion due to the reduced water depth. While the impact of bed slope on wave propagation celerity is documented, it is often overlooked in commonly used depth-integrated wave models. This study uses the WKB approximation and solves higher-order slope-related terms to analyze the influence of varying depth, including the gradient and laplacian of water depth. One result is an extended longwave for linear wave reflection and transmission on a ramp to deeper waters. The main outcome and focus of this work, however, is a new, simple analytical expression for linear dispersion that includes bed variations. The results are applied to two cases: wind-generated water wave propagation in the nearshore, emphasizing corrections to bathymetry inversion methods, and tsunami propagation over the continental slope, highlighting the limitations of neglecting slope on dispersion and the significant role of the Laplacian of water depth.

Relaxation rate of ModMax–de Sitter black holes perturbed by massless neutral scalar fields

In this study, we investigate the behavior of ModMax-de Sitter black holes when perturbed by massless neutral scalar fields. Specifically, we analyze how the relaxation time, defined as the inverse of the fundamental imaginary frequency, varies with respect to two key parameters: the cosmological constant and the nonlinear parameter characterizing the ModMax theory. We explore scenarios both with and without a cosmological constant, focusing on the static charged ModMax black hole configuration. Our results reveal dependencies between the relaxation time and the nonlinear parameter, shedding light on the dynamical properties of these black hole systems. We also show the validity of WKB approximation under consideration.

Love wave propagation in a smart composite structure of linear and exponential functionally graded porous piezoelectric material

Functionally graded materials (FGMs) have emerged as a promising avenue for enhancing the performance and longevity of surface acoustic wave (SAW) devices. This importance is gained due to gradual variation in functional properties of FGMs. In this paper, a mathematical model of a piezoelectric layer overlying a functionally graded porous piezoelectric (FGPP) layer resting on the elastic substrate is presented for the study of Love waves. The material parameters of the FGPP layer are considered to vary linearly and exponentially with the thickness of the layer. The solutions of governing equations corresponding to the FGPP layer are obtained by the Wentzel Kramers Brillouin (WKB) method. Dispersion equations are derived for both electrically open and shorted boundaries, linear and exponential gradation in both mechanical and electrical parameters, and for gradation in mechanical parameters alone as well. After numerical computation, dispersion curves are plotted to investigate the influence of wavenumber, type of gradation, extent of gradation, and type of boundaries on the phase and group velocity. The phase velocity decreases with wavenumber and is found more for linear gradation in comparison to exponential gradation. The electromechanical coupling factor is analyzed for different propagating modes of Love waves, in the case when gradation in mechanical properties is considered, the electromechanical coupling factor is greater than that when variation in both mechanical and electrical properties is considered. It is also observed that the tailoring of gradation can help to get the suitable value of the electromechanical coupling factor. The insights obtained from these findings can offer valuable contributions toward advancing the functionality and efficiency of SAW devices, there by bolstering their applicability in various technological domains.

An effective model for the quantum Schwarzschild black hole: Weak deflection angle, quasinormal modes and bounding of greybody factor

In this paper, we thoroughly explore two crucial aspects of a quantum Schwarzschild black solution within four-dimensional space–time: (i) the weak deflection angle, (ii) the rigorous greybody factor and, (iii) the Dirac quasinormal modes. Our investigation involves employing the Gauss–Bonnet theorem to precisely compute the deflection angle and establishing its correlation with the Einstein ring. Additionally, we derive the rigorous bounds for greybody factors through the utilization of general bounds for reflection and transmission coefficients in the context of Schrodinger-like one-dimensional potential scattering. We also compute the corresponding Dirac quasinormal modes using the WKB approximation. We reduce the Dirac equation to a Schrodinger-like differential equation and solve it with appropriate boundary conditions to obtain the quasinormal frequencies. To visually underscore the quantum effect, we present figures that illustrate the impact of varying the parameter r0, or more specifically, in terms of the parameter α. This comprehensive examination enhances our understanding of the quantum characteristics inherent in the Schwarzschild black solution, shedding light on both the deflection angle and greybody factors in a four-dimensional space–time framework.

Probing the black holes in a dark matter halo of M87 using gravitational wave echoes

Variations at the event horizon structure of a black hole will emit the signals of the gravitational wave echoes associated with the ringdown of the binary black hole merger. In this work, combining mass model of M87 and Einasto profile for dark matter halo, we construct one formal solution of black holes in dark matter halo, and this solution includes the regular black hole for geometric parameter a<2M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a < 2M$$\\end{document} and the wormhole for a≥2M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a \\ge 2M$$\\end{document}. Then, we test this solution under axial gravitational perturbation and calculate their quasinormal modes (QNM). Our results show that when geometric parameter a>2M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a>2M$$\\end{document}, a series of gravitational wave echoes appear after the QNM, and one distinctive feature of gravitational wave echoes is the double barrier. Besides, we also study the impacts of shape parameters α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} of Einasto profile both on the QNM and gravitational wave echoes, and extract their frequencies using WKB method and Prony method. In principle, with the increasing of shape parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}, the frequencies of the QNM and gravitational wave echoes between the different shape parameters must be different. But now there is zero difference of the frequencies between shape parameter α=0.18\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =0.18$$\\end{document} and α=0.20\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =0.20$$\\end{document}. Zero difference is not allowed, so we give an upper limit for shape parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}, which is approximately α<0.18\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha <0.18$$\\end{document}. These signals of gravitational wave echoes in a dark matter halo may be detected in the near future, and these characteristics of gravitational wave echoes may serve as local measurements of dark matter.

Optimally truncated WKB approximation for the 1D stationary Schrödinger equation in the highly oscillatory regime

This paper is dedicated to the efficient numerical computation of solutions to the 1D stationary Schrödinger equation in the highly oscillatory regime. We compute an approximate solution based on the well-known WKB-ansatz, which relies on an asymptotic expansion w.r.t. the small parameter ɛ. Assuming that the coefficient in the equation is analytic, we derive an explicit error estimate for the truncated WKB series, in terms of ɛ and the truncation order N. For any fixed ɛ, this allows to determine the optimal truncation order Nopt which turns out to be proportional to ɛ−1. When chosen this way, the resulting error of the optimally truncated WKB series behaves like O(exp(−r/ɛ)), with some parameter r>0. The theoretical results established in this paper are confirmed by several numerical examples.

Maxwell's equal area law for Vaidya-Bonner-de Sitter black hole under Lorentz invariance violation

In this study, we investigate the tunneling of fermions with arbitrary spin near the event horizon of a nonstationary Vaidya-Bonner-de Sitter (VBdS) black hole under Lorentz invariance violation (LIV). The modified Hawking temperature of VBdS black holes is calculated by using tortoise coordinate transformation, Feynman prescription, and Wentzel–Kramers–Brillouin approximation. By considering the cosmological constant as a thermodynamic pressure in the extended phase space, we construct a Maxwell's equal area law under LIV and study the phase transitions of VBdS black hole in , , and planes. The LIV increases the length of the liquid-gas coexistence region. The thermodynamic quantities such as the entropy, heat capacity, Helmholtz free energy, internal energy, enthalpy, and Gibbs free energy of the VBdS black hole are discussed. These quantities tend to increase under LIV. The stability of the black hole is also discussed in the presence of LIV.

Effects of non-commutative geometry on black hole properties

In this study, we investigate the signatures of a non-commutative black hole solution. Initially, we calculate the thermodynamic properties of the system, including entropy, heat capacity, and Hawking radiation. For the latter quantity, we employ two distinct methods: surface gravity and the topological approach. Additionally, we examine the emission rate and remnant mass within this context. Remarkably, the lifetime of the black hole, after reaching its final state due to the evaporation process, is expressed analytically up to a grey-body factor. We estimate the lifetime for specific initial and final mass configurations. Also, we analyze the tensorial quasinormal modes using the 6th-order WKB method. Finally, we study the deflection angle, i.e., gravitational lensing, in both the weak and strong deflection limits.

Quasinormal mode of Schwarzschild black hole with geometric correction

We investigated the quasinormal modes (QNMs) for scalar, Dirac and electromagnetic fields of a Schwarzschild black hole with geometric corrections. In addition to the inherent modes of the Schwarzschild black hole, we discovered the existence of purely imaginary modes induced by the geometric corrections, which dominated the behavior of the perturbation when geometric corrections were pretty small. Notably, we found that these purely imaginary modes are prominent, and their imaginary parts are proportional to the correction parameter ϵ. We also explored the large-l limit using the WKB approximation for scalar field. This opens up the possibility of observed geometric corrections in the future.

Quasinormal modes of the Schwarzschild black hole with a deficit solid angle and quintessence-like matter: Improved asymptotic iteration method

We study the quasinormal modes (QNM) for scalar, and electromagnetic perturbations in the Schwarzschild black hole with a deficit solid angle and quintessence-like matter. Using the sixth--order WKB approximation and the improved asymptotic iteration method (AIM) we can determine the dependence of the quasinormal modes on the parameters of the black hole and the parameters on the test fields. The values of the real part and imaginary parts of the quasi--normal modes increase with the decrease of the values of the deficit solid angle and density of quintessence-like matter. The quasinormal modes gotten by these two methods are in good agreement. Using the finite difference method, we obtain the time evolution profile of such perturbations in this Black Hole.

Dymnikova black hole from an infinite tower of higher-curvature corrections

Recently, in [1], it was demonstrated that various regular black hole metrics can be derived within a theory featuring an infinite number of higher curvature corrections to General Relativity. Moreover, truncating this infinite series at the first few orders already yields a reliable approximation of the observable characteristics of such black holes [2]. Here, we further establish the existence of another regular black hole solution, particularly the D-dimensional extension of the Dymnikova black hole, within the equations of motion incorporating an infinite tower of higher-curvature corrections. This solution is essentially nonperturbative in the coupling parameter, rendering the action, if it exists, incapable of being approximated by a finite number of powers of the curvature. In addition, we compute the dominant quasinormal frequencies of such black holes using both the Bernstein polynomial method and the 13th order WKB method with Padé approximants, obtaining a high degree of agreement between them.

Higher-Dimensional Communications Using Multimode Fibers and Compact Components to Enable a Dense Set of Communicating Channels

Higher-dimensional communications are of interest for multiple reasons, including increasing the classical transmission capacity and, more recently, the quantum state transfer through fibers using the many modes within the fiber. For quantum communications, this enables an increase in the number of bits per photon, increasing quantum fidelity, increasing error thresholds and enabling hyperentanglement transfer, among other possibilities. A high-dimensional quantum state transfer can be transported through multimode fiber using the many modes available. However, this transfer of information through multimode optical fiber is limited by attenuation and mode coupling among the various spatial and polarization modes. Here, we consider how this mode coupling impacts the transfer process. We consider the fiber’s modal properties, including orbital angular momentum, modal group numbers, and principal modes. We also investigate and propose input and output optical components, as well as fiber properties, which better mitigate the deleterious effects of mode coupling. We use the WKB approximation to the scaler wave equation as a guidance to quantify this coupling and then implement corrections to this approximation using exact solutions to the scaler wave equation. We consider methods to circumvent this mode coupling using optical fiber designs, holographic optical components and devices that are commercially available today. Some of these components, such as the holographic gratings and lenses, could be implemented using flat optics.

Massless Dirac perturbations of black holes in f(Q) gravity: quasinormal modes and a weak deflection angle

This article considers a static and spherical black hole (BH) in f(Q) gravity. f(Q) gravity is the extension of symmetric teleparallel general relativity, where both curvature and torsion are vanishing and gravity is described by nonmetricity. In this study, we investigate the possible implications of quasinormal mode (QNM) modified Hawking spectra and deflection angles generated by the model. The Wentzel–Kramers–Brillouin method is used to solve the equations of motion for massless Dirac perturbation fields and explore the impact of the nonmetricity parameter (Q 0). Based on the QNM computation, we can ensure that the BH is stable against massless Dirac perturbations and as Q 0 increases the oscillatory frequency of the mode decreases. We then discuss the weak deflection angle in the weak field limit approximation. We compute the deflection angle up to the fourth order of approximation and show how the nonmetricity parameter affects it. We find that the Q 0 parameter reduces the deflection angle.