Radial Part Research Articles

$\mathrm{BC}_{2}$-Type Multivariable Matrix Functions and Matrix Spherical Functions

Matrix spherical functions associated to the compact symmetric pair (\mathrm{SU}(m+2), \mathrm{S}(\mathrm{U}(2)\times \mathrm{U}(m))) , m\geq 2 , having a reduced root system of type \mathrm{BC}_{2} , are studied. We consider an irreducible K -representation (\pi,V) arising from the \mathrm{U}(2) -part of K , and the induced representation \mathrm{Ind}_{K}^{G} \pi splits multiplicity-free. The corresponding spherical functions, i.e. \Phi \colon G \to \mathrm{End}(V) satisfying \Phi(k_{1}gk_{2})=\pi(k_{1})\Phi(g)\pi(k_{2}) for all g\in G , k_{1},k_{2}\in K , are studied by examining certain leading terms which involve hypergeometric functions. This is done explicitly using the action of the radial part of the Casimir operator on these functions and their leading terms. To suitably grouped matrix spherical functions we associate two-variable matrix orthogonal polynomials giving a matrix analogue of Koornwinder’s 1970s two-variable orthogonal polynomials, which are Heckman–Opdam polynomials for \mathrm{BC}_{2} . In particular, we find explicit orthogonality relations with the matrix polynomials being eigenfunctions to an explicit second-order matrix partial differential operator. The scalar part of the matrix weight is less general than Koornwinder’s weight.

Vibration-rotational alternating-parity spectra of even–even nuclei with effective triaxiality

In the present work, a model combining vibration-rotational motion and effective triaxiality is developed for even–even nuclei with quadrupole and octupole deformations. The Davidson potential is used to solve the radial part of the vibration-rotational Schrödinger equation in the axial quadrupole and octupole variables. In an adiabatic approximation the contribution of the angular variable in the components of the moments of inertia is taken as a constant after which the triaxial-rotor energy is obtained. As a result the alternating-parity spectrum and wave functions of the nucleus with quadrupole-octupole degrees of freedom are obtained. The proposed model is used to describe the yrast and first-non-yrast alternating-parity bands of the even–even nuclei 150Nd, 154Sm, 154,160Gd, 156Dy, 162,164Er, 172Yb, 230,232Th, 230,232,234,236,238U and 238,240Pu. The relevance of the model for taking into account the collective quadrupole-octupole modes in a rather complete form is pointed out.

Massive Dirac equation in static Bumblebee black hole space-times, detailed derivations and novel exact solutions

In this work, we construct and investigate the relativistic spin-12 fermionic fields quantum dynamics in static spherically symmetric Bumblebee black hole background. The derivation of the Dirac equation in a general static spherically symmetric black hole space-time is carried out in detail via tetrad formalism. With the help of total angular momentum operator, the angular equation can be separated from the radial part where the solution is given in terms of the spinor harmonics. The radial Dirac equation has a well-known problem due to the presence of the square root terms appearing simultaneously with the rest mass that prevents us to find exact solutions. In this work, we present exact solutions of light mass fermion's wave function and energy levels bound in the static Bumblebee black hole. We discover the exact solutions of the massive Dirac's radial equation in terms of the Confluent Heun functions. Moreover, thanks to the well-known polynomial condition of the Confluent Heun functions, we also derive the energy quantization.

Relativistic scalar fields canonical quantization in Einstein–Yang–Mills–Higgs’s rotating black hole space-time

The quantum theory of relativistic mechanics to deal with the scalar fields behavior in a curved space-time is represented by the Klein–Gordon equation. In this work, we will investigate the gravitationally bound states of massive and massless scalar fields around a Einstein–Yang–Mills–Higgs’s rotating black hole. After applying the standard separation of variables ansatz, we will show in detail how to obtain the novel exact solutions of the radial part of the governing Klein–Gordon equation and express the radial solution in terms of the Confluent Heun functions. By applying the bound state boundary conditions, the Confluent Heun functions are reduced to be polynomials that lead to energy quantization. We find that the scalar fields have complex-valued energy levels that indicate the decaying/growing bound states known as quasibound states. In the end, using the exact radial solution, we derive the radiation distribution function of the black hole’s outer horizon to obtain the equation of the Hawking temperature.

Exact massless spinor quasibound states of Schwarzschild black hole

In this work, we investigate the behavior of massless spin 12 field by working out the Dirac equation in a curved static spherically symmetric Schwarzschild space-time. Detailed derivation of novel exact massive and massless scalar quasibound state in Schwarzschild black hole background is presented in this work. After decoupling the Dirac equation, we successfully solve both of the angular and radial parts, respectively in terms of spin weighted Spherical Harmonics and Confluent Heun functions. With the exact radial wave solution in hand, applying its polynomial condition leads to the discovery of the quantized energy levels expression. We found that the massless spinor field around the Schwarzschild black hole has complex valued energy levels, in contrast to the purely imaginary for a massless boson around the same black hole. In the last section, the Hawking radiation distribution function is derived via Damour-Ruffini method and the Hawking temperature is obtained.

Fuzzy onionlike space as a matrix model

We propose a matrix model realization of a three-dimensional quantum space. It has an onionlike structure composed of concentric fuzzy spheres of increasing radius. The angular part of the Laplace operator is inherited from that of the fuzzy sphere. The radial part is constructed using operators that relate matrices of various sizes using the matrix harmonic expansion. As an example of this approach, we produce a numerical simulation of a scalar quantum field theory, the classical heat transfer, study the quantum-mechanical hydrogen atom, and consider some analytical aspects of the scalar field theory on this space. Published by the American Physical Society 2024

Variation of leaf turgor and pressure parameters evaluation in drip-irrigated apple canopy

Leaf turgor pressure is a relevant metric for determining plant water content. To diagnose an apple tree water shortage, it is critical to precisely establish the features of leaf turgor pressure change in the canopy. In the current study, we continually monitored the leaf turgor pressure parameter (Pp) at different azimuthal, vertical, and radial locations of the apple canopy based on leaf patch clamp pressure (LPCP) probes. The results demonstrated that the Pp curves on sunny and rainy days were significantly different. A peak occurred on sunny days, during which the turgor pressure remained constant around midday. Under full irrigation, there was a greater change rate and daily maximum of Pp (maxPp) at the east (azimuthal), upper layer (vertical), and distal (radial) positions. While leaf turgor pressure dramatically decreased with incomplete watering, maxPp significantly increased at the central and lower layers of the east canopy, as well as on the proximal and middle of the radial parts. This spatial heterogeneity in canopy leaf turgor pressure was related to the soil water potential, midday leaf water potential, sponge tissue thickness, and canopy solar radiation. Then, the ideal drought stress indicator parameter (maxPp) and canopy turgor monitoring leaf location (the middle position of the eastern canopy central layer's radial branch) were determined by multiparameter assessment. The findings presented above serve as a theoretical foundation for the construction of an intelligent irrigation system based on apple tree water status monitoring.

A numerical Poisson solver with improved radial solutions for a self-consistent locally scaled self-interaction correction method

The universal applicability of density functional approximations is limited by self-interaction error made by these functionals. Recently, a novel one-electron self-interaction-correction (SIC) method that uses an iso-orbital indicator to apply the SIC at each point in space by scaling the exchange-correlation and Coulomb energy densities was proposed. The locally scaled SIC (LSIC) method is exact for the one-electron densities, and unlike the well-known Perdew–Zunger SIC (PZSIC) method recovers the uniform electron gas limit of the uncorrected density functional approximation, and reduces to PZSIC method as a special case when isoorbital indicator is set to the unity. Here, we present a numerical scheme that we have adopted to evaluate the Coulomb potential of the electron density scaled by the iso-orbital indicator required for the self-consistent LSIC calculations. After analyzing the behavior of the finite difference method (FDM) and the green function solution to the radial part of the Poisson equation, we adopt a hybrid approach that uses the FDM for the Coulomb potential due to the monopole and the GF for all higher-order terms. The performance of the resultant hybrid method is assessed using a variety of systems. The results show improved accuracy than earlier numerical schemes. We also find that, even with a generic set of radial grid parameters, accurate energy differences can be obtained using a numerical Coulomb solver in standard density functional studies.

Analytical Solution of Non-Linear Oscillator and Thermal Properties of Interacting System of Real Polyatomic Molecules

The study examined the solution of nonlinear oscillator and thermal properties of interacting system of real polyatomic molecules. A nonlinear oscillator potential was substituted into the radial part of the Schrodinger wave equation and solved using the Frobenius method to obtain the energy eigenvalues of the system. The partition function and the energy eigenvalues are used to determine three thermal properties for four polyatomic molecules which showed that added energy is required in the absence of the nonlinear term as the orbital and magnetic quantum numbers are increased and degeneracy completely removed. Other findings also reveal that as the virial coefficients of the four polyatomic molecules vary, the additive property of entropy is observed as well as the enhancement of the mean energy approximately proportional to the volume of the molecules. The four polyatomic molecules under consideration are environmentally impactful and are indispensable to man.

Effective interaction between guest charges immersed in 2D jellium

The model under study is an infinite 2D jellium of pointlike particles with elementary charge e, interacting via the logarithmic potential and in thermal equilibrium at the inverse temperature β. Two cases of the coupling constant Γ≡βe2 are considered: the Debye–Hückel limit Γ→0 and the free-fermion point Γ=2 . In the most general formulation, two guest particles, the one with charge qe (the valence q being an arbitrary integer) and the hard core of radius σ > 0 and the pointlike one with elementary charge e, are immersed in the bulk of the jellium at distance d⩾σ . Two problems are of interest: the asymptotic large-distance behavior of the excess charge density induced in the jellium and the effective interaction between the guest particles. Technically, the induced charge density and the effective interaction are expressed in terms of multi-particle correlations of the pure (translationally invariant) jellium system. It is shown that the separation form of the induced charge density onto its radial and angle parts, observed previously in the limit Γ→0 , is not reproduced at the coupling Γ=2 . Based on an exact expression for the effective interaction between guest particles at Γ=2 , oppositely ( q=0,−1,−2,… ) charged guest particles always attract one another while likely ( q=1,2,… ) charged guest particles repeal one another up to a certain distance d between them and then the mutual attraction takes place up to asymptotically large (finite) distances.

Vacuum energy of scalar fields on spherical shells with general matching conditions

In this work we analyze the spectral zeta function for massless scalar fields propagating in a D-dimensional flat space under the influence of a shell potential. The static nature of the potential, and the spherical symmetry, allows us to focus on the spatial part of the field which satisfies a one-dimensional Schrodinger equation endowed with a point potential. The shell potential is defined in terms of the two-interval self-adjoint extensions of the one-dimensional Schrodinger equation that describes the radial part of the scalar field. After performing the necessary analytic continuation, we utilize the spectral zeta function of the system to compute the vacuum energy of the field.

Scattering theory and an index theorem on the radial part of SL(2, ℝ)

We present the spectral and scattering theory of the Casimir operator acting on radial functions in [Formula: see text]. After a suitable decomposition, these investigations consist in studying a family of differential operators acting on the half-line. For these operators, explicit expressions can be found for the resolvent, for the spectral density, and for the Møller wave operators, in terms of the Gauss hypergeometric function. An index theorem is also introduced and discussed. The resulting equality, generically called Levinson’s theorem, links various asymptotic behaviors of the hypergeometric function. This work is a first attempt to connect group theory, special functions, scattering theory, [Formula: see text]-algebras, and Levinson’s theorem.

Numerical solution of the Schrödinger equation using Neural Networks in Python

The motion of quantum mechanical systems in physical sciences is described by partial differential equations, usually of second order with respect to spatial coordinates. The required solutions of the time-dependent type of equations are, in general, functions of the temporal variable t and the state vectors determining the positions of the system’s particles at the time t. Only a small number, however of those differential equations can however be solved analytically, while the majority of them must be solved numerically by applying specific very advanced integration techniques. Among these equations, the fundamental Schrödinger equation offers great insight towards numerical solving. In this work we present an effective method that solves numerically the time-independent Schrodinger equation on the basis of neural networks techniques. Analytical and numerical results for the radial part of this equation with the corresponding energies are then compared so as to estimate the performance of our method.

Investigation of hidden symmetry in AdS5 black hole with Heun equation

Investigating the existence of algebra and finding hidden symmetries in physical systems is one of the most important aspects for understanding their behavior and predicting their future. Expanding this unique method of study to cosmic structures and combining past knowledge with new data can be very interesting and lead to discovering new ways to analyze these systems. However, studying black hole symmetries always presents many complications and sometimes requires computational approximations. For example, checking the existence of Killing vectors and then calculating them is not always an easy task. It becomes much more difficult as the structure and geometry of the system become more complex. In this work, we will show that if the wave equations with a black hole background can be converted in the form of general Heun equation, based on its structure and coefficients, the algebra of the system can be easily studied, and computational and geometrical complications can be omitted. For this purpose, we selected two [Formula: see text] black holes: Reissner–Nordstrm (RN) and Kerr, and analyzed the Klein–Gordon equation with the background of these black holes. Based on this concept, we observed that the radial part of the RN black hole and both the radial and angular parts of the Kerr black hole could be transformed into the general form of the Heun equation. As a result, according to the algebraic structure that governs the Heun equation and its coefficients, one can easily achieve generalized sl(2) algebra.

ηc production within light-cone approach in pp and eA collisions

Abstract. The charmonium state ηc is a pseudoscalar meson (JPC = 0−+). We will review the proton-proton and electron-ion reactions, where ηc can be produced through the fusion of two virtual gluons in pp collision or one virtual photon-one real photon in eA collision. The main ingredient in our light-cone approach is the space-like transition form factor with dependence on two virtu-alities of the particles. The idea is to construct the form factor based on the wave function of the cc¯ state. The radial part of the wave function can be found by solving the Schrödinger equation. The transformation to the light-cone variable is performed using the Terentev prescription. We present the effects of the so-obtained form factors in the context of electron-ion collisions. Future facilities such as EIC or LHeC give the opportunity to probe the form factor dependence on the virtuality in e−A collisions. Our results indicate that the electron-ion colliders can be considered as an alternative and provide supplementary data to those obtained in e+e− colliders.

Electron plasma diagnostics in ELTRAP by electron cyclotron resonance heating method.

Electron cyclotron resonance heating method of Particle-in-Cell code was used to analyze heating phenomena, axial kinetic energy, and self-consistent electric field of confined electron plasma in ELTRAP device by hydrogen and helium background gases. The electromagnetic simulations were performed at a constant power of 3.8 V for different RF drives (0.5 GHz- 8 GHz), as well as for 1 GHz constant frequency at these varying amplitudes (1 V-3.8 V). The impacts of axial and radial temperatures were found maximum at 1.8 V and 5 GHz as compared to other amplitudes and frequencies for both background gases. These effects are higher at varying radio frequencies due to more ionization and secondary electrons production and maximum recorded radial temperature for hydrogen background gas was 170.41 eV. The axial kinetic energy impacts were found more effective in the outer radial part (between 0.03 and 0.04 meters) of the ELTRAP device due to applied VRF through C8 electrode. The self-consistent electric field was found higher for helium background gas at 5 GHz RF than other amplitudes and radio frequencies. The excitation and ionization rates were found to be higher along the radial direction (r-axis) than the axial direction (z-axis) in helium background gas as compared to hydrogen background gas. The current studies are advantageous for nuclear physics applications, beam physics, microelectronics, coherent radiation devices and also in magnetrons.

From Scalar Clouds to Rotating Hairy Black Holes

First, we review the solutions of a complex-valued scalar field, termed scalar clouds, with and without electric charge, when coupled to a rotating Kerr–Newman (electrically charged) or Kerr (neutral) black hole (BH), respectively. To this aim, we determine the conditions and parameters that characterize the existence of solutions that represent bound states, with an energy-momentum tensor that respect the symmetries of the underlying spacetimes, even if the backreaction of the field is not taken into account at this stage. In particular, we show that in the extremal Kerr scenario the cloud solutions exist only when the mass of the BH satisfies certain bounds, which are obtained by analyzing an effective potential associated with the radial dependency of the scalar clouds that leads to a Schrödinger-like equation. Second, when the backreaction of the field in the spacetime is taken into account, we present a family of stationary, axisymmetric and asymptotically flat solutions of the Einstein–Klein–Gordon system that represent genuine rotating hairy black holes (RHBHs) and provide different values of some global quantities associated with them, such as the Komar mass and the Komar angular momentum. We also compute RHBH solutions with nodes in the radial part of the scalar field and also for a higher azimuthal number m.

Open-Source Instrumented Object to Study Dexterous Object Manipulation.

Humans use tactile feedback to perform skillful manipulation. When tactile sensory feedback is unavailable, for instance, if the fingers are anesthetized, dexterity is severely impaired. Imaging the deformation of the finger pad skin when in contact with a transparent plate provides information about the tactile feedback received by the central nervous system. Indeed, skin deformations are transduced into neural signals by the mechanoreceptors of the finger pad skin. Understanding how this feedback is used for active object manipulation would improve our understanding of human dexterity. In this paper, we present a new device for imaging the skin of the finger pad of one finger during manipulation performed with a precision grip. The device's mass (300 g) makes it easy to use during unconstrained dexterous manipulation. Using this device, we reproduced the experiment performed in Delhaye et al. (2021) We extracted the strains aligned with the object's movement, i.e., the vertical strains in the ulnar and radial parts of the fingerpad, to see how correlated they were with the grip force (GF) adaptation. Interestingly, parts of our results differed from those in Delhaye et al. (2021) due to weight and inertia differences between the devices, with average GF across participants differing significantly. Our results highlight a large variability in the behavior of the skin across participants, with generally low correlations between strain and GF adjustments, suggesting that skin deformations are not the primary driver of GF adaptation in this manipulation scenario.

Solving exact solution of time independent Schrodinger wave equation and visualizing eigenfunctions using XCOS

Time independent Schrodinger wave equation (SWE) for a quantum system can be solved computationally which involves application of numerical method. Different computational tools used to solve SWE involving different potentials were reviewed. XCOS, an inbuilt tool of Scilab for modelling and simulating hybrid dynamic system and is an open-source is used to solve the Schrodinger wave equation for different quantum systems. Only basic understanding of mathematics up to high school level is required to solve Schrodinger wave equation using XCOS. Different quantum systems considered are free particle in a well bounded by infinite potential at the boundaries, radial part of SWE for Hydrogen atom under coulombic potential, screened coulombic potential. Radial part of Schrodinger wave equation for a particle under harmonic and anharmonic potential were also solved using XCOS computational tool. Eigenvalues and eigenvectors were obtained by solving radial part of Schrodinger wave equation for molecules under Morse potential. Advantages and limitation of XCOS computational tools were also discussed. We argue in favour of the academic use of computational tool XCOS to solve Schrodinger wave equation in introductory quantum mechanics courses.

[formula omitted]-norm spherical copulas

In this paper we study Lp-norm spherical copulas for arbitrary p∈[1,∞] and arbitrary dimensions. The study is motivated by a conjecture that these distributions lead to a sharp bound for the value of a certain generalized mean difference. We fully characterize conditions for existence and uniqueness of Lp-norm spherical copulas. Explicit formulas for their densities and correlation coefficients are derived and the distribution of the radial part is determined. Moreover, statistical inference and efficient simulation are considered.