Radial Eigenfunctions Research Articles

Sheer shear: weak lensing with one mode

3D data compression techniques can be used to determine the natural basis of radial eigenmodes that encode the maximum amount of information in a tomographic large-scale structure survey. We explore the potential of the Karhunen-Lo\`eve decomposition in reducing the dimensionality of the data vector for cosmic shear measurements, and apply it to the final data from the \cfh survey. We find that practically all of the cosmological information can be encoded in one single radial eigenmode, from which we are able to reproduce compatible constraints with those found in the fiducial tomographic analysis (done with 7 redshift bins) with a factor of ~30 fewer datapoints. This simplifies the problem of computing the two-point function covariance matrix from mock catalogues by the same factor, or by a factor of ~800 for an analytical covariance. The resulting set of radial eigenfunctions is close to ell-independent, and therefore they can be used as redshift-dependent galaxy weights. This simplifies the application of the Karhunen-Lo\`eve decomposition to real-space and Fourier-space data, and allows one to explore the effective radial window function of the principal eigenmodes as well as the associated shear maps in order to identify potential systematics. We also apply the method to extended parameter spaces and verify that additional information may be gained by including a second mode to break parameter degeneracies. The data and analysis code are publicly available at https://github.com/emiliobellini/kl_sample.

Applying quantum supersymmetry and perturbation theory to the energy-dependent Hulthén potential

We deal with the energy-dependent Hulthen potential, by using the supersymmetric quantum mechanics and the first-order perturbation theory. We consider the Hulthen potential linearly dependent on the energy which is introduced in the coupling constant. We evaluate the energy eigenvalues and the corresponding reduced radial eigenfunctions. We compare the analytical results with the numerical solutions of the Schrodinger equation in the energy-dependent and the energy-independent cases. We then investigate the screening effect in order to determine the critical screening parameter values for each state. After that, we look into the absorption oscillator strengths for different transitions.

Relativistic spectral bounds for the general molecular potential: application to a diatomic molecule.

We tackle with the Dirac equation in the presence of the general molecular potential (GMP). The spin symmetric solution of the relativistic wave equation is obtained by considering the Pekeris-type approximation scheme to deal with the centrifugal term, and in order to solve second-order differential equation, the asymptotic iteration method (AIM) is used. The closed form of the energy eigenvalue equation is found out for any values of the angular momentum quantum number. We calculate the relativistic vibrational bound state energies of 51Δg state of Na2 molecule and compare them with the Rydberg-Klein-Rees (RKR) data. We show that relativistic vibrational energies of this molecule, which are found in the spin symmetric case, are more convenient with experimental RKR data than non-relativistic vibrational energies. We also obtain normalization constant by considering inductive approach and investigate the radial eigenfunctions and probability density functions corresponding to different eigenvalues graphically for the Na2(51Δg) molecule.

The reproducing kernel of $\mathcal H^2$ and radial eigenfunctions of the hyperbolic Laplacian

In the paper we characterize the reproducing kernel $\mathcal {K}_{n,h}$ for the Hardy space $\mathcal {H}^2$ of hyperbolic harmonic functions on the unit ball $\mathbb {B}$ in $\mathbb {R}^n$. Specifically we prove that \[ \mathcal {K}_{n,h}(x,y) = \sum _{\alpha =0}^\infty S_{n,\alpha }(\lvert x\rvert )S_{n,\alpha }(\lvert y\rvert ) Z_\alpha (x,y), \] where the series converges absolutely and uniformly on $K\times \mathbb {B}$ for every compact subset $K$ of $\mathbb {B}$. In the above, $S_{n,\alpha }$ is a hypergeometric function and $Z_\alpha $ is the reproducing kernel of the space of spherical harmonics of degree α. In the paper we prove that \[ 0\le \mathcal K_{n,h}(x,y) \le \frac {C_n}{(1-2\langle x,y\rangle + \lvert x \rvert^2 \lvert y \rvert^2)^{n-1}}, \] where $C_n$ is a constant depending only on $n$. It is known that the diagonal function $\mathcal K_{n,h}(x,x)$ is a radial eigenfunction of the hyperbolic Laplacian $\varDelta_h $ on $\mathbb{B} $ with eigenvalue $\lambda _2 = 8(n-1)^2$. The result for $n=4$ provides motivation that leads to an explicit characterization of all radial eigenfunctions of $\varDelta_h $ on $\mathbb{B} $. Specifically, if $g$ is a radial eigenfunction of $\varDelta_h $ with eigenvalue $\lambda _\alpha = 4(n-1)^2\alpha (\alpha -1)$, then \[ g(r) = g(0) \frac {p_{n,\alpha }(r^2)}{(1-r^2)^{(\alpha -1)(n-1)}}, \] where $p_{n,\alpha }$ is again a hypergeometric function. If α is an integer, then $p_{n,\alpha }(r^2)$ is a polynomial of degree $2(\alpha -1)(n-1)$.

Localized spin waves in isolated kπ skyrmions

The localized magnon modes of isolated $k\pi$ skyrmions on a field-polarized background are analyzed based on the Landau-Lifshitz-Gilbert equation within the terms of an atomistic classical spin model, with system parameters based on the Pd/Fe biatomic layer on Ir(111). For increasing skyrmion order $k$ a higher number of excitation modes are found, including modes with nodes in the radial eigenfunctions. It is shown that at low fields $2\pi$ and $3\pi$ skyrmions are destroyed via a burst instability connected to a breathing mode, while $1\pi$ skyrmions undergo an elliptic instability. At high fields all $k\pi$ skyrmions collapse due to the instability of a breathing mode. The effective damping parameters of the spin waves are calculated in the low Gilbert damping limit, and they are found to diverge in the case of the lowest-lying modes at the burst and collapse instabilities, but not at the elliptic instability. It is shown that the breathing modes of $k\pi$ skyrmions may become overdamped at higher Gilbert damping values.

On the Helical MHD Mode in a Finite-Conductivity Tokamak Plasma

A helical MHD perturbation in a finite-conductivity tokamak plasma has been considered in the straight-cylinder model in a situation where there is no resonance surface q = m/n in plasma. The radial eigenfunction of the helical mode, in addition to the large-scale component described at σ||→ ∞ by the ideal MHD equation, contains a small-scale component localized near the wall and near discontinuities in the radial profiles of the unperturbed quantities. At smooth profiles, the small-scale component is attached to the wall and is smaller in magnitude than the large-scale component. Therefore, beyond a thin near-wall plasma layer, the mode is close to the large-scale ideal MHD mode. The presence of the small-scale component is necessary to satisfy the boundary conditions for the perturbed field on the wall.

Characterizing the harmonic manifolds by the eigenfunctions of the Laplacian

The space forms, the complex hyperbolic spaces and the quaternionic hyperbolic spaces are characterized as the harmonic manifolds with specific radial eigenfunctions of the Laplacian.

From the nonrelativistic Morse potential to a unified treatment of a large class of bound‐state solutions of a modified D ‐dimensional Klein–Gordon equation

From the nonrelativistic Morse potential to a unified treatment of a large class of bound‐state solutions of a modified <i>D</i> ‐dimensional Klein–Gordon equation

Non-Relativistic Treatment of a Generalized Inverse Quadratic Yukawa Potential

A bound state solution is a quantum state solution of a particle subjected to a potential such that the particle’s energy is less than the potential at both negative and positive infinity. The particle’s energy may also be negative as the potential approaches zero at infinity. It is characterized by the discretized eigenvalues and eigenfunctions, which contain all the necessary information regarding the quantum systems under consideration. The bound state problems need to be extended using a more precise method and approximation scheme. This study focuses on the non-relativistic bound state solutions to the generalized inverse quadratic Yukawa potential. The expression for the non-relativistic energy eigenvalues and radial eigenfunctions are derived using proper quantization rule and formula method, respectively. The results reveal that both the ground and first excited energy eigenvalues depend largely on the angular momentum numbers, screening parameters, reduced mass, and the potential depth. The energy eigenvalues, angular momentum numbers, screening parameters, reduced mass, and the potential depth or potential coupling strength determine the nature of bound state of quantum particles. The explored model is also suitable for explaining both the bound and continuum states of quantum systems.

Controllability of a 2D quantum particle in a time-varying disc with radial data

In this article we consider a 2-D quantum particle confined in a disc whose radius can be deformed continuously in time. We study the problem of controllability of such a quantum particle via deformations of the initial disc, i.e., when we set the time-dependent radius of the disc to be the control variable. We prove that the resulting system is locally controllable around some radial trajectories which are linear combinations of the first three radial eigenfunctions of the Laplacian in the unit disc with Dirichlet boundary conditions. We prove this result, thanks to the linearisation principle, by studying the linearised system, which leads to a moment problem that can be solved using some results from Nonharmonic Fourier series. In particular, we have to deal with fine properties of Bessel functions.

Quasi-bound state resonances of charged massive scalar fields in the near-extremal Reissner\u2013Nordstr\xf6m black-hole spacetime

The quasi-bound states of charged massive scalar fields in the near-extremal charged Reissner–Nordström black-hole spacetime are studied analytically. These discrete resonant modes of the composed black-hole-field system are characterized by the physically motivated boundary condition of ingoing waves at the black-hole horizon and exponentially decaying (bounded) radial eigenfunctions at spatial infinity. Solving the Klein–Gordon wave equation for the linearized scalar fields in the black-hole spacetime, we derive a remarkably compact analytical formula for the complex frequency spectrum which characterizes the quasi-bound state resonances of the composed Reissner–Nordström-black-hole-charged-massive-scalar-field system.

On some unexpected properties of radial and symmetric eigenvalues and eigenfunctions of the p-Laplacian on a disk

We discuss several properties of eigenvalues and eigenfunctions of the p-Laplacian on a ball subject to zero Dirichlet boundary conditions. Among main results, in two dimensions, we show the existence of nonradial eigenfunctions which correspond to the radial eigenvalues. Also we prove the existence of eigenfunctions whose shape of the nodal set cannot occur in the linear case p=2. Moreover, the limit behavior of some eigenvalues as p→1+ and p→+∞ is studied.

Accidental Degeneracy of Hydrogen Atom Revisited

We examine accidental symmetry of hydrogen-like atoms analytically in spherical coordinates. Some unfamiliar but notable functional relations of the radial eigenfunctions of system’s Hamiltonian are constructed, for both discrete and continuous energy eigenvalues. Analytical similarity of the symmetry between two-dimensional and threedimensional hydrogen atoms are therefore demonstrated.

The effect of Jupiter oscillations on Juno gravity measurements

Seismology represents a unique method to probe the interiors of giant planets. Recently, Saturn’s f-modes have been indirectly observed in its rings, and there is strong evidence for the detection of Jupiter global modes by means of ground-based, spatially-resolved, velocimetry measurements. We propose to exploit Juno’s extremely accurate radio science data by looking at the gravity perturbations that Jupiter’s acoustic modes would produce. We evaluate the perturbation to Jupiter’s gravitational field using the oscillation spectrum of a polytrope with index 1 and the corresponding radial eigenfunctions. We show that Juno will be most sensitive to the fundamental mode (n=0), unless its amplitude is smaller than 0.5 cm/s, i.e. 100 times weaker than the n∼4−11 modes detected by spatially-resolved velocimetry. The oscillations yield contributions to Juno’s measured gravitational coefficients similar to or larger than those expected from shallow zonal winds (extending to depths less than 300 km). In the case of a strong f-mode (radial velocity ∼ 30 cm/s), these contributions would become of the same order as those expected from deep zonal winds (extending to 3000 km), especially on the low degree zonal harmonics, therefore requiring a new approach to the analysis of Juno data.

From the generalized Morse potential to a unified treatment of the D-dimensional singular harmonic oscillator and singular Coulomb potentials

Bound-state solutions of the singular harmonic oscillator and singular Coulomb potentials in arbitrary dimensions are generated in a simple way from the solutions of the one-dimensional generalized Morse potential. The nonsingular harmonic oscillator and nonsingular Coulomb potentials in arbitrary dimensions with their additional accidental degeneracies are obtained as particular cases. Added bonuses from these mappings are the straightforward determination of the critical attractive singular potential, the proper boundary condition on the radial eigenfunction at the origin and the inexistence of bound states in a pure inversely quadratic potential.

On the shielding of sound from a source near one or two coaxial cones

The shielding of the sound from a source near a cone or two coaxial circular cones is addressed by an exact analytical method, considering free spherical waves satisfying a rigid or impedance boundary condition on the wall of the cone(s); the boundary condition is applied for impedance proportional to the distance from the vertex. The exact solution involves the eigenvalues of the problem that are generally real or complex (not integers), and coincide with the degree of the associated Legendre functions and order of the spherical Bessel functions which specify respectively the latitudinal and radial dependencies of the wave field. These eigenfunctions or ‘conical wave harmonics’ can be chosen in more than one way, e.g. (a) as standing wave modes which are finite at the vertex of the cone(s), but do not satisfy a radiation condition at infinity; (b) as propagating waves satisfying a radiation condition at infinity, and generally singular at the vertex of the cone(s). The method of calculation of eigenvalues and eigenfunctions is presented both for a single cone and two coaxial cones with the same vertex, and arbitrary aperture(s). The method specifies the eigenvalues, and the corresponding radial, azimuthal and latitudinal eigenfunctions. An asymptotic formula is obtained for the eigenvalues which gives reasonable good agreement with the exact results. The eigenvalues and eigenfunctions appear with suitable amplitudes in the Green function representing a monopole source near the vertex of the cone(s). The acoustic field is plotted also for a longitudinal and a transverse dipole and mixed quadrupole source near the vertex.

An analog of the Schwartz theorem on spectral analysis on a hyperbolic plane

Let \( \mathbb{D} \) be an open unit disk in the complex plane. It is shown that every subspace in C(\( \mathbb{D} \)) invariant under weighted conformal shifts contains a radial eigenfunction of the corresponding invariant differential operator. This function can be expressed via the Gauss hypergeometric function and is a generalization of the spherical function on the disk \( \mathbb{D} \) which is considered as a hyperbolic plane with the corresponding Riemannian structure.

Resistive interchange mode destabilized by helically trapped energetic ions and its effects on energetic ions and bulk plasma in a helical plasma

A resistive interchange mode of the structure (, : poloidal and toroidal mode numbers, respectively) with a bursting character and rapid frequency chirping in the range less than 10 kHz is observed for the first time in the edge region of the net current-free, low beta LHD (Large Helical Device) plasmas during high power injection of perpendicular neutral beams. The mode resonates with the precession motion of helically trapped energetic ions (EPs), following the resonant condition. The radial mode structure is recognized to be similar to that of the pressure-driven resistive interchange mode, of which radial displacement eigenfunction quite localizes around the mode rational surface, and evolves into an odd-type (or island-type) during the late of frequency chirping phase. This beam driven mode is excited when the beta value of helically trapped EPs exceeds a certain threshold. This instability is thought to be a new branch of resistive interchange mode destabilized by the trapped energetic ions. The radial transport, i.e. redistribution and losses, of helically trapped energetic ions induced by the mode transiently generates significant radial electric field near the plasma peripheral region. The large shear of thus generated radial electric field is thought to contribute to the observed suppression of micro-turbulence and transient increases of the temperature of fully ionized carbon impurity ions and electron density, suggesting improvement of bulk plasma confinement.

Analytical solution of scattering by a DB/D′B′ cylinder coated with multilayer anisotropic media for cloaking applications

In this study, analytical solution of plane wave scattering is presented by a circular DB/D′B′ cylinder coated with multilayer anisotropic media. The relative permittivity and permeability tensors of each layer, when referring to principle axes (ρ, φ, z), are biaxial and diagonal; so, the radial eigenfunctions are complex ordered Bessel's functions. In order to compare different boundary conditions and cloaks, scattering width is calculated in some examples. Theory and simulation results show that scattering width of the circular DB (or D′B′) cylinder coated with Cloak I (or Cloak II) has the same value in both TMz and TEz polarisations. Furthermore, it is demonstrated that the DB boundary condition and Cloak I have good performance in the invisibility of cylindrical structures, which may provide a potential way to design a dual polarised cloak.

Rossby waves in an azimuthal wind

Rossby waves in an azimuthal wind are analyzed using an eigen-function expansion. Solutions of the wave equation for the stream-function for Rossby waves are obtained in which depends on where r is the cylindrical radius, is the azimuthal angle measured in the plane relative to the Easterly direction, (the -plane is locally horizontal to the Earth’s surface in which the x-axis points East, and the y-axis points North). The radial eigenfunctions in the -plane are Bessel functions of order and argument kr, where k is a characteristic wave number and have the form in which the satisfy recurrence relations involving , , and . The recurrence relations for the have solutions in terms of Bessel functions of order where is the frequency of the wave and is the angular velocity of the wind and argument . By summing the Bessel function series, the complete solution for reduces to a single Bessel function of the first kind of order . The argument of the Bessel function is a complicated expression depending on r, , a, and kr. These solutions of the Rossby wave equation can be interpreted as being due to wave-wave interactions in a locally rotating wind about the local vertical direction. The physical characteristics of the rotating wind Rossby waves are investigated in the long and short wavelength limits; in the limit as the azimuthal wind velocity ; and in the zero frequency limit in which one obtains a stationary spatial pattern for the waves. The vorticity structure of the waves are investigated. Time dependent solutions with are also investigated.