1,608 research outputs found
An infinite family of superintegrable Hamiltonians with reflection in the plane
We introduce a new infinite class of superintegrable quantum systems in the
plane. Their Hamiltonians involve reflection operators. The associated
Schr\"odinger equations admit separation of variables in polar coordinates and
are exactly solvable. The angular part of the wave function is expressed in
terms of little -1 Jacobi polynomials. The spectra exhibit "accidental"
degeneracies. The superintegrability of the model is proved using the
recurrence relation approach. The (higher-order) constants of motion are
constructed and the structure equations of the symmetry algebra obtained.Comment: 19 page
The Dunkl oscillator in the plane I : superintegrability, separated wavefunctions and overlap coefficients
The isotropic Dunkl oscillator model in the plane is investigated. The model
is defined by a Hamiltonian constructed from the combination of two independent
parabosonic oscillators. The system is superintegrable and its symmetry
generators are obtained by the Schwinger construction using parabosonic
creation/annihilation operators. The algebra generated by the constants of
motion, which we term the Schwinger-Dunkl algebra, is an extension of the Lie
algebra u(2) with involutions. The system admits separation of variables in
both Cartesian and polar coordinates. The separated wavefunctions are
respectively expressed in terms of generalized Hermite polynomials and products
of Jacobi and Laguerre polynomials. Moreover, the so-called Jacobi-Dunkl
polynomials appear as eigenfunctions of the symmetry operator responsible for
the separation of variables in polar coordinates. The expansion coefficients
between the Cartesian and polar bases (overlap coefficients) are given as
linear combinations of dual -1 Hahn polynomials. The connection with the
Clebsch-Gordan problem of the sl_{-1}(2) algebra is explained.Comment: 25 pages; Added references; Added appendix on anti-Hermicity of the
Dunkl derivativ
Generalized squeezed-coherent states of the finite one-dimensional oscillator and matrix multi-orthogonality
A set of generalized squeezed-coherent states for the finite u(2) oscillator
is obtained. These states are given as linear combinations of the mode
eigenstates with amplitudes determined by matrix elements of exponentials in
the su(2) generators. These matrix elements are given in the (N+1)-dimensional
basis of the finite oscillator eigenstates and are seen to involve 3x3 matrix
multi-orthogonal polynomials Q_n(k) in a discrete variable k which have the
Krawtchouk and vector-orthogonal polynomials as their building blocks. The
algebraic setting allows for the characterization of these polynomials and the
computation of mean values in the squeezed-coherent states. In the limit where
N goes to infinity and the discrete oscillator approaches the standard harmonic
oscillator, the polynomials tend to 2x2 matrix orthogonal polynomials and the
squeezed-coherent states tend to those of the standard oscillator.Comment: 18 pages, 1 figur
The Dunkl oscillator in the plane II : representations of the symmetry algebra
The superintegrability, wavefunctions and overlap coefficients of the Dunkl
oscillator model in the plane were considered in the first part. Here
finite-dimensional representations of the symmetry algebra of the system,
called the Schwinger-Dunkl algebra sd(2), are investigated. The algebra sd(2)
has six generators, including two involutions and a central element, and can be
seen as a deformation of the Lie algebra u(2). Two of the symmetry generators,
J_3 and J_2, are respectively associated to the separation of variables in
Cartesian and polar coordinates. Using the parabosonic creation/annihilation
operators, two bases for the representations of sd(2), the Cartesian and
circular bases, are constructed. In the Cartesian basis, the operator J_3 is
diagonal and the operator J_2 acts in a tridiagonal fashion. In the circular
basis, the operator J_2 is block upper-triangular with all blocks 2x2 and the
operator J_3 acts in a tridiagonal fashion. The expansion coefficients between
the two bases are given by the Krawtchouk polynomials. In the general case, the
eigenvectors of J_2 in the circular basis are generated by the Heun polynomials
and their components are expressed in terms of the para-Krawtchouk polynomials.
In the fully isotropic case, the eigenvectors of J_2 are generated by little -1
Jacobi or ordinary Jacobi polynomials. The basis in which the operator J_2 is
diagonal is then considered. In this basis, the defining relations of the
Schwinger-Dunkl algebra imply that J_3 acts in a block tridiagonal fashion with
all blocks 2x2. The matrix elements of J_3 in this basis are given explicitly.Comment: 33 page
Tridiagonalization of the hypergeometric operator and the Racah-Wilson algebra
The algebraic underpinning of the tridiagonalization procedure is
investigated. The focus is put on the tridiagonalization of the hypergeometric
operator and its associated quadratic Jacobi algebra. It is shown that under
tridiagonalization, the quadratic Jacobi algebra becomes the quadratic
Racah-Wilson algebra associated to the generic Racah/Wilson polynomials. A
degenerate case leading to the Hahn algebra is also discussed.Comment: 14 pages; Section 3 revise
An Algebraic Model for the Multiple Meixner Polynomials of the First Kind
An interpretation of the multiple Meixner polynomials of the first kind is
provided through an infinite Lie algebra realized in terms of the creation and
annihilation operators of a set of independent oscillators. The model is used
to derive properties of these orthogonal polynomials
The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl_{-1}(2)
The algebra H of the dual -1 Hahn polynomials is derived and shown to arise
in the Clebsch-Gordan problem of sl_{-1}(2). The dual -1 Hahn polynomials are
the bispectral polynomials of a discrete argument obtained from a q-> -1 limit
of the dual q-Hahn polynomials. The Hopf algebra sl_{-1}(2) has four generators
including an involution, it is also a q-> -1 limit of the quantum algebra
sl_{q}(2) and furthermore, the dynamical algebra of the parabose oscillator.
The algebra H, a two-parameter generalization of u(2) with an involution as
additional generator, is first derived from the recurrence relation of the -1
Hahn polynomials. It is then shown that H can be realized in terms of the
generators of two added sl_{-1}(2) algebras, so that the Clebsch-Gordan
coefficients of sl_{-1}(2) are dual -1 Hahn polynomials. An irreducible
representation of H involving five-diagonal matrices and connected to the
difference equation of the dual -1 Hahn polynomials is constructed.Comment: 15 pages, Some minor changes from version #
LISACode : A scientific simulator of LISA
A new LISA simulator (LISACode) is presented. Its ambition is to achieve a
new degree of sophistication allowing to map, as closely as possible, the
impact of the different sub-systems on the measurements. LISACode is not a
detailed simulator at the engineering level but rather a tool whose purpose is
to bridge the gap between the basic principles of LISA and a future,
sophisticated end-to-end simulator. This is achieved by introducing, in a
realistic manner, most of the ingredients that will influence LISA's
sensitivity as well as the application of TDI combinations. Many user-defined
parameters allow the code to study different configurations of LISA thus
helping to finalize the definition of the detector. Another important use of
LISACode is in generating time series for data analysis developments
The Dynamics of Sustained Reentry in a Loop Model with Discrete Gap Junction Resistance
Dynamics of reentry are studied in a one dimensional loop of model cardiac
cells with discrete intercellular gap junction resistance (). Each cell is
represented by a continuous cable with ionic current given by a modified
Beeler-Reuter formulation. For below a limiting value, propagation is found
to change from period-1 to quasi-periodic () at a critical loop length
() that decreases with . Quasi-periodic reentry exists from
to a minimum length () that is also shortening with .
The decrease of is not a simple scaling, but the bifurcation can
still be predicted from the slope of the restitution curve giving the duration
of the action potential as a function of the diastolic interval. However, the
shape of the restitution curve changes with .Comment: 6 pages, 7 figure
Ab initio study of the thermodynamic properties of rare-earthmagnesium intermetallics MgRE (RE=Y, Dy, Pr, Tb)
We have performed an ab initio study of the thermodynamical properties of
rare-earth-magnesium intermetallic compounds MgRE (RE=Y, Dy, Pr, Tb) with
CsCl-type B2-type structures. The calculations have been carried out the
density functional theory and density functional perturbation theory in
combination with the quasiharmonic approximation. The phonon-dispersion curves
and phonon total and partial density of states have been investigated. Our
results show that the contribution of RE atoms is dominant in phonon frequency,
and this character agrees with the previous discussion by using atomistic
simulations. The temperature dependence of various quantities such as the
thermal expansions, bulk modulus, and the heat capacity are obtained. The
electronic contributions to the specific heat are discussed, and found to be
important for the calculated MgRE intermetallics.Comment: 12 pages, 6 figure
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