Radial Eigenfunctions Research Articles

Supersymmetry and the Hartmann potential of theoretical chemistry

An exactly solvable ring-shaped potential in quantum chemistry given by $$V = \eta \sigma ^2 \varepsilon _{\text{o}} \left( {\frac{{2a_{\text{O}} }}{r} - \frac{{\eta a_{\text{O}}^2 }}{{r^2 {\text{sin}}^{\text{2}} \theta }}} \right)$$ was introduced by Hartmann in 1972 to describe ring-shaped molecules like benzene. In this article, the supersymmetric features of the Hartmann potential are discussed, We first review the results of a previous paper in which we rederived the eigenvalues and radial eigenfunctions of the Hartmann potential using a formulation of one-dimensional supersymmetric quantum mechanics (SUSYQM) on the half-line [0, ∞). A reformulation of SUSYQM in the full line (− ∞, ∞) is subsequently developed. It is found that the second formulation makes a connection between states having the same quantum number L but different values of ησ2 and quantum number N. This is in contrast to the first formulation, which relates states with identical values of the quantum number N and ησ2 but different values of the quantum number L.

A Line-of-Sight Integration Approach to Cosmic Microwave Background Anisotropies

We present a new method for calculating linear cosmic microwave background (CMB) anisotropy spectra based on integration over sources along the photon past light cone. In this approach the temperature anisotropy is written as a time integral over the product of a geometrical term and a source term. The geometrical term is given by radial eigenfunctions which do not depend on the particular cosmological model. The source term can be expressed in terms of photon, baryon and metric perturbations, all of which can be calculated using a small number of differential equations. This split clearly separates between the dynamical and geometrical effects on the CMB anisotropies. More importantly, it allows to significantly reduce the computational time compared to standard methods. This is achieved because the source term, which depends on the model and is generally the most time consuming part of calculation, is a slowly varying function of wavelength and needs to be evaluated only in a small number of points. The geometrical term, which oscillates much more rapidly than the source term, does not depend on the particular model and can be precomputed in advance. Standard methods that do not separate the two terms and require a much higher number of evaluations. The new method leads to about two orders of magnitude reduction in CPU time when compared to standard methods and typically requires a few minutes on a workstation for a single model. The method should be especially useful for accurate determinations of cosmological parameters from CMB anisotropy and polarization measurements that will become possible with the next generation of experiments. A programm implementing this method can be obtained from the authors.

Supersymmetric treatment of a particle subjected to a ring-shaped potential

The ring-shaped Hartmann potential V = ησǫ0 ( 2a0 r − ηa 0 rsinθ ) was introduced in quantum chemistry to describe ring-shaped molecules like benzene. In this article, fundamental concepts of supersymmetric quantum mechanics (SUSYQM) are discussed. The energy eigenvalues and (radial) eigenfunctions of the Hartmann potential are subsequently rederived using the techniques of SUSYQM.

Supersymmetry and the Hartmann potential of theoretical chemistry

The ring-shaped Hartmann potential $ V = \eta \sigma^{2} \epsilon_{0} \left( \frac{2 a_{0}}{r} - \frac{\eta a_{0}^{2}}{r^{2} sin^{2} \theta} \right) $ was introduced in quantum chemistry to describe ring-shaped molecules like benzene. In this article, the supersymmetric features of the Hartmann potential are discussed. We first review the results of a previous paper in which we rederived the eigenvalues and radial eigenfunctions of the Hartmann potential using a formulation of one-dimensional supersymmetric quantum mechanics (SUSYQM) on the half-line $\left[ 0, \infty \right)$. A reformulation of SUSYQM in the full line $\left( -\infty, \infty \right)$ is subsequently developed. It is found that the second formulation makes a connection between states having the same quantum number $L$ but different values of $\eta \sigma^{2}$ and quantum number $N$. This is in contrast to the first formulation, which relates states with identical values of the quantum number $N$ and $\eta \sigma^{2}$ but different values of the quantum number $L$.

Force-free thin flux tubes: Basic equations and stability

The thin flux tube approximation is considered for a straight, symmetrical, force-free, rigidly rotating flux tube. The derived set of equations describes tube, body sausage, and Alfvén wave modes and is valid for any values of β. The linear waves and instabilities of force-free flux tubes are considered. The comparison of approximate and exact solutions for an untwisted, nonrotating flux tube is performed. It is shown that the approximate and exact dispersion equations coincides, except the 20% discrepancy of sausage frequencies. An effective cross section is proposed to introduce the removal of this discrepancy. It makes the derived approximation correct for the force-free thin flux tube dynamics, except the detailed structure of radial eigenfunction. The dispersion of Alfvén torsional waves in a force-free tubes appears. The valve effect of one directional propagation of waves in rotating twisted tube is revealed. The current and rotational sausage instabilities of a force-free, thin flux tube are considered.

Streamline topology in eccentric Taylor vortex flow

We investigate an asymptotic model of DiPrima & Stuart (1972b, 1975) describing steady Taylor vortex flow between eccentric cylinders, under the assumption that the eccentricity ε, the clearance ratio δ and the Taylor vortex amplitude A satisfy ε, δ and A small. By solving a boundary value problem for the radial eigenfunctions we numerically obtain the flow field of DiPrima & Stuart and investigate its topology, after correcting higher-order terms to ensure that the flow preserves volume. We find regions of chaotic streamlines at all eccentricities and discuss the reason for their existence. We make an analogy between the full model and a modulated vortex flow field which qualitatively displays the same behaviour. For large eccentricities, we examine the flow field and the topology of its streamlines, especially where the two-dimensional flow contains a separated region of recirculation. In this case Taylor vortices give rise to transport of fluid particles in and out of the separated region. We find that the onset of Taylor vortices encourages recirculation in the inflow plane, whilst discouraging it in the outflow plane.

On the global stability of magnetized accretion disks. 1: Axisymmetric modes

view Abstract Citations (40) References (28) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS On the Global Stability of Magnetized Accretion Disks. I. Axisymmetric Modes Curry, C. ; Pudritz, R. E. ; Sutherland, Peter G. Abstract We investigate the global stability of differentially rotating fluid shell threaded by a constant vertical magnetic field to linear, axisymmetric perturbations. Such a system is known to be unstable due to the interaction between the magnetic field and fluid shear, and should mimic the behavior of an accretion disk far from its vertical boundaries. The magnetic field exterior to the fluid has an effect on the radial boundary motion, and we derive the appropiate boundary conditions. The growth rates of the instability, associated radial eigenfunctions, and critical field strengths for stability are presented for a wide variety of radii, rotation profiles, and magnetic field strengths. The growth rates of unstable global modes are always less than, but comparable to, the corresponding local growth rates. For near critical field strengths, nonrigid boundary conditions lead to much higher growth rates than do (physically unrealistic) rigid boundary conditions. Also, we find that the critical Alfven speed for stability is large (of order the sound speed) for realistic disk sizes, thus implying that the instability is always present. Finally, the connection between the global and local characters of the instability is elucidated. Publication: The Astrophysical Journal Pub Date: October 1994 DOI: 10.1086/174718 Bibcode: 1994ApJ...434..206C Keywords: Accretion Disks; Axisymmetric Flow; Boundary Layers; Equations Of State; Interstellar Matter; Magnetic Fields; Numerical Analysis; Perturbation Theory; Angular Momentum; Eigenvalues; Kepler Laws; Magnetohydrodynamics; Rotary Stability; Astrophysics; ACCRETION; ACCRETION DISKS; INSTABILITIES; ISM: MAGNETIC FIELDS; MAGNETOHYDRODYNAMICS: MHD full text sources ADS | Related Materials (1) Part 2: 1995ApJ...453..697C

EM-wave propagation through semi-elliptic cylindrical dielectric waveguide on a perfectly conductive planar substrate

This is a theoretical study of modes in a semielliptic cylindrical dielectric waveguide on a perfectly conductive planar substrate. Interest is mainly in high eccentricity, electrically large, hence multimode, structures. The theory applies to light propagation through surface-wave transmission lines or microwave ducting by shore-bounded evaporation ducts. The rigorous formulation of fields in terms of radial and azimuthal Mathieu eigenfunctions is followed by accurate computations of the longitudinal phase constant, the fractional power trapped within the waveguide and the group velocity. The effects of size, ellipticity and refractive index step on mode features are investigated. As information, previously unavailable, about high-order modes is reported, an interesting classification of modes readily unfolds.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Surface-Wave Propagation On A Rotating, Anisotropic Earth

SUMMARY We present a JWKB theory for the propagation of monochromatic surface waves on a rotating, anisotropic, laterally heterogeneous earth model. the theory allows for slowly varying topography on the earth's surface and any internal discontinuity, and incorporates the effect of self-gravitation and anelasticity on the wavefield. the analysis is based upon slowly varying variational principles and involves a local dispersion relation and local radial eigenfunctions which depend explicitly on the direction of the local wavevector as a result of the rotation and anisotropy of the earth model. the amplitude is determined by a conservation law for the surface-wave energy. In addition to the usual dynamical phase, which is the integral of the local wavevector along a ray path, there is an additional variation in phase. All rotating earth models, isotropic models included, support such an additional variation in phase, which is an analogue of the Berry phase in adiabatic quantum mechanics. the Berry phase vanishes in non-rotating earth models with a local radial two-fold axis or a local horizontal mirror plane.

A calculation with a biorthogonal wavelet transformation

The use of a biorthogonal basis for continuous wavelet transformations is explored, thus relaxing the so-called admissibility condition on the analyzing wavelet. As an application, the eigenvalues and corresponding radial eigenfunctions of the Hamiltonian of relativistic hydrogen-like atoms are determined.

Unsteady Lifting Surface Theory for Supersonic Through-Flow Fan.

The purpose of the present study is to predict analytically the unsteady loading on vibrating blades of a supersonic through-flow fan and investigate the aerodynamic instability. It is assumed that each blade operates with zero steady loading, and vibrates with small displacement amplitude, and a linearized unsteady lifting surface theory for a supersonic through-flow fan on the basis of the finite radial eigenfunction series approximation is developed. Numerical results of the unsteady loadings and unsteady aerodynamic works are presented. The bending vibration with zero mean loading is always stable, and the torsional instability heavily depends upon relative locations of the leading edge Mach wave reflection point and the torsion axis.

Variational Principles For Surface Wave Propagation On A Laterally Heterogeneous Earth-Iii. Potential Representation

Summary We present a frequency-domain analysis of travelling and standing surface waves on a smooth, laterally heterogeneous Earth model, using a potential representation that is valid everywhere, including in the neighbourhood of surface wave caustics. the Love and Rayleigh wave displacement fields are written in the form uL=k-1LW (-řX▿1)χL and uR=UřχR+k-1RV▿1χR; the quantities U, V and W are the local radial eigenfunctions, kL and kR are the local Love and Rayleigh wavenumbers, and χL and χR are surface wave potentials that vary rapidly on the surface of the unit sphere. the natural normalization condition for the local radial eigenfunctions is cCI1= 1, where c is the local phase velocity, C is the local group velocity and I1 is the local radial kinetic energy integral; with this normalization, the Love and Rayleigh wave potentials satisfy the spherical Helmholtz equations ▿21χL + k2LχL = 0 and ▿21χR + k2RχR = 0. Standing wave eigenfunctions χL and χR can be determined either by solving a truncated matrix eigenvalue problem or by using the EBK semi-classical method; the results incorporate multiplet coupling along a single dispersion branch but ignore cross-branch coupling. the theory allows for slowly varying topography of the Earth's surface and the core-mantle boundary, and incorporates the effect of self-gravitation.

Inversion of experimental data and ab initio studies of a pseudo-atom–diatom model for the vibrational dynamics of HCN–HF

Spectroscopic data was inverted to generate simplified atom–diatom intermolecular potentials for the hydrogen bonded dimer HCN–HF. Both the HF ν1=0 and ν1=1 adiabatic surfaces of this complex involving the hydrogen bond stretch (ν4) and the high frequency HF bend (ν6) degrees of freedom were considered. Adiabatic separation of angular and radial degrees of freedom allowed a modified vibrational Rydberg–Klein–Rees (RKR) inversion to give effective radial potentials for each bending state. The final potential at a sequence of intermolecular separations was obtained by requiring agreement between the eigenvalues of the angular Hamiltonian and the effective radial potentials which were obtained from the RKR inversion. Potentials obtained from the inversion procedure were tested by a variational calculation with a basis set that consisted of products of preoptimized radial and angular eigenfunctions. Transition frequencies included in the inversion procedure were reproduced to better than 1 cm−1, respective rotational constants were predicted within 0.15% of the experimental value and predicted intensities were in qualitative agreement with experimental results. Ab initio potentials were also calculated using second order Mo/ller–Plesset perturbation theory to treat electron correlation effects and with a triple-zeta-valence basis set plus two sets of polarization functions. The energies were computed at geometries where the potential had been determined by the inversion procedure. A total of 264 geometries were considered. A correction for basis set superposition errors led to good agreement between ab initio and experimental values of the well depth. We calculated the bound states of the ab initio surfaces assuming the adiabatic separation of the ν1 mode from the ν4 and the ν6 modes. The transition frequencies calculated from the ab initio surface differed from the experimental energies by less than 20 cm−1 even for highly excited overtones. Both the potentials obtained from the inversion procedure and from the ab initio calculations were modified to predict the spectrum of HCN–DF.

Performance Analysis of Supersonic Through-Flow Fan by the Lifting Surface Theory. 1st Report. Disturbance Flow Field and Determination of Blade Loadings.

The purpose of the present study, which is composed of two parts, is to predict analytically the steady performance of a supersonic through-flow fan in terms of design parameters. It is assumed that the fan cascade operates at supersnic axial velocity with a small angle of attack, camber and thickness. In 1st Report, the linearized lifting surface theory for a rotating annular cascade on the basis of the finite radial eigenfunction series approximation developed by the first author is extended to the supersonic through-flow fan model. Expressions of the disturbance flow field are derived, and the integral equation for the steady loading is numerically solved. The fundamental features of the blade surface pressure distribution dependent on the blade profile are demonstrated. The extensive parametric study of steady performance is presented in 2nd Report.

Unsteady Lifting Surface Theory for Supersonic Through-Flow Fan.

The purpose of this work is to predict analytically the unsteady loading on vibrating blades of a supersonic through-flow fan and investigate the aerodynamic instability. In order to deal with this problem, a linearized unsteady lifting surface theory on the basis of the finite radial eigenfunction series approxmation is developed. It is assumed that each blade operates with zero mean loading, and vibrates with a small displacement amplitude. Numerical results for pure bending, pure torsional and combined bending and torsional vibrations are presented to demonstrate influences of interblade phase angle, location of torsion axis, axial Mach number and reduced frequency on the flutter boundary.

Theory of nonlinear optics in dielectric spheres II Coupled-partial-wave theory of resonant, resonantly pumped stimulated Brillouin scattering

I derive a self-consistent-field, coupled-mode theory of the generation of stimulated Brillouin scattering (SBS) in transparent dielectric spheres by using full vector-partial-wave theory. The results are expressed in an analytical form that is suitable for efficient numerical evaluation. The theoretical method introduced in this paper uses the Racah–Wigner angular-momentum calculus to derive selection rules for the allowed nonlinear couplings of partial waves and uses Fourier–Bessel and Fourier–Dini expansions for the radial eigenfunctions of the electromagnetic and acoustic normal modes. The derivation shows that the SBS field must be resonant in order to satisfy the electromagnetic boundary conditions.

Drift wave instability in the Io plasma torus

We present a linear normal‐mode analysis of low‐frequency electrostatic drift waves in the Io plasma torus, providing a kinetic description of the centrifugal interchange instability of the torus regulated by coupling to Jupiter's ionosphere. We assume a periodic potential perturbation with azimuthal wavenumber m≫1 and solve a boundary value problem to obtain the wave dispersion relation and radial eigenfunction. For given m the growing mode is a standing wave in the corotating reference frame. If the outer torus boundary is taken as a discontinuity, the growth rate is proportional to m times the torus flux tube content times the ion drift frequency, divided by the Pedersen conductivity of Jupiter's ionosphere. The inner torus boundary has a modest stabilizing effect for azimuthal wavelengths greater than the radial thickness of the torus. The finite slope of the outer torus density profile has a more pronounced stabilizing effect, reducing the growth rate by the factor 2β/m, where β∼2 is the exponent of the assumed power law decline of flux tube content with distance. Even so, the e‐folding time is of the order of 1 hour, much less than the inferred residence time of torus ions. The growth rate can be further reduced dramatically by the stabilizing effect of an as‐yet unobserved ring current distribution, the “impoundment” effect proposed by Siscoe et al. (1981). Various theoretical models of global radial transport in Jupiter's magnetosphere, including corotating convection, interchange diffusion, and transient flux tube convection, can be understood as plausible nonlinear evolutions of electrostatic drift waves. Further observations and/or numerical simulations are needed to ascertain the relative importance of these transport mechanisms.

Effects of a transition zone on free oscillations of the Earth

Reference Earth models have internal boundaries across which elastic parameters are discontinuous. Frequencies of free oscillations are perturbed if the abrupt boundary is replaced by a transition zone in which elastic parameters change gradually. The frequency perturbation is explicitly expressed in terms of radial eigenfunctions of reference models and elastic parameters in the transition zone. The effects may be observable for the inner core boundary.

On the evaluation of the integral over the product of two spherical Bessel functions

The integral Il,l′(k,k′)=∫∞0jl (kr)jl′(k′r)r 2 dr, in which the spherical Bessel functions jl(kr) are the radial eigenfunctions of the three-dimensional wave equation in spherical coordinates, is evaluated in terms of distributions, in particular, step functions and delta functions. It will be shown that the behavior of Il,l′ is very different in the cases l−l′ even (0, ±2, ±4, ...) and l−l′ odd (±1, ±3, ...). For l−l′ even it is expressed in terms of the delta function, step functions, and Legendre polynomials. For l−l′ odd it is expressed in terms of Legendre functions of the second kind and step functions; no delta functions appear.

Asymptotic behaviour of solutions of elliptic equations with critical exponents and Neumann boundary conditions

SynopsisAsymptotic estimates are established for nontrivial positive radial eigenfunctions of the nonlinear eigenvalue problem −Δu= λ(up−uq) in the unit ballBin ℝN(N&gt; 2) with Neumann boundary conditions, as the supremum norm tends to infinity. Herepis the critical Sobolev exponent (N+ 2)/(N− 2) and 0 &lt;q&lt;p− 1 = 4/(N− 2).