329 research outputs found
Confluent hypergeometric orthogonal polynomials related to the rational quantum Calogero system with harmonic confinement
Two families (type and type ) of confluent hypergeometric polynomials
in several variables are studied. We describe the orthogonality properties,
differential equations, and Pieri type recurrence formulas for these families.
In the one-variable case, the polynomials in question reduce to the Hermite
polynomials (type ) and the Laguerre polynomials (type ), respectively.
The multivariable confluent hypergeometric families considered here may be used
to diagonalize the rational quantum Calogero models with harmonic confinement
(for the classical root systems) and are closely connected to the (symmetric)
generalized spherical harmonics investigated by Dunkl.Comment: AMS-LaTeX v1.2 (with amssymb.sty), 34 page
The Direct Effect of Toroidal Magnetic Fields on Stellar Oscillations: An Analytical Expression for the General Matrix Element
Where is the solar dynamo located and what is its modus operandi? These are
still open questions in solar physics. Helio- and asteroseismology can help
answer them by enabling us to study solar and stellar internal structures
through global oscillations. The properties of solar and stellar acoustic modes
are changing with the level of magnetic activity. However, until now, the
inference on subsurface magnetic fields with seismic measures has been very
limited. The aim of this paper is to develop a formalism to calculate the
effect of large-scale toroidal magnetic fields on solar and stellar global
oscillation eigenfunctions and eigenfrequencies. If the Lorentz force is added
to the equilibrium equation of motion, stellar eigenmodes can couple. In
quasi-degenerate perturbation theory, this coupling, also known as the direct
effect, can be quantified by the general matrix element. We present the
analytical expression of the matrix element for a superposition of subsurface
zonal toroidal magnetic field configurations. The matrix element is important
for forward calculations of perturbed solar and stellar eigenfunctions and
frequency perturbations. The results presented here will help to ascertain
solar and stellar large-scale subsurface magnetic fields, and their geometric
configuration, strength, and their change over the course of activity cycles.Comment: 20 pages, accepted for publication in The Astrophysical Journa
On the oscillation of the laterally heterogeneous earth, 1
The perturbative effects, as cause by lateral inhomogeneities in the earth structure and by Coriolis force, contaminate the originally toroidal and spheroidal earth's oscillations, making them of mixed type. For this reason, in order to make the computation of the perturbations more uniform and homogeneous, it was suggested that the earth's free oscillations be expanded into a series in terms of generalized harmonics familiar from the theory of angular momentum in quantum mechanics. Making use of Gibbsian symbolism and of some operators from the theory of angular momentum, explicit expressions were deduced for the perturbative terms in the differential equation of the earth's free oscillations. Decomposition of the strain tensor in terms of canonical vectors was also obtained
Generalized Ellipsoidal and Sphero-Conal Harmonics
Classical ellipsoidal and sphero-conal harmonics are polynomial solutions of
the Laplace equation that can be expressed in terms of Lame polynomials.
Generalized ellipsoidal and sphero-conal harmonics are polynomial solutions of
the more general Dunkl equation that can be expressed in terms of Stieltjes
polynomials. Niven's formula connecting ellipsoidal and sphero-conal harmonics
is generalized. Moreover, generalized ellipsoidal harmonics are applied to
solve the Dirichlet problem for Dunkl's equation on ellipsoids.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Local inverse scattering at fixed energy in spherically symmetric asymptotically hyperbolic manifolds
In this paper, we adapt the well-known \emph{local} uniqueness results of
Borg-Marchenko type in the inverse problems for one dimensional Schr{\"o}dinger
equation to prove \emph{local} uniqueness results in the setting of inverse
\emph{metric} problems. More specifically, we consider a class of spherically
symmetric manifolds having two asymptotically hyperbolic ends and study the
scattering properties of massless Dirac waves evolving on such manifolds. Using
the spherical symmetry of the model, the stationary scattering is encoded by a
countable family of one-dimensional Dirac equations. This allows us to define
the corresponding transmission coefficients and reflection
coefficients and of a Dirac wave having a fixed
energy and angular momentum . For instance, the reflection
coefficients correspond to the scattering experiment in which a
wave is sent from the \emph{left} end in the remote past and measured in the
same left end in the future. The main result of this paper is an inverse
uniqueness result local in nature. Namely, we prove that for a fixed , the knowledge of the reflection coefficients (resp.
) - up to a precise error term of the form with
B\textgreater{}0 - determines the manifold in a neighbourhood of the left
(resp. right) end, the size of this neighbourhood depending on the magnitude
of the error term. The crucial ingredients in the proof of this result are
the Complex Angular Momentum method as well as some useful uniqueness results
for Laplace transforms.Comment: 24 page
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