810 research outputs found
Upward extension of the Jacobi matrix for orthogonal polynomials
Orthogonal polynomials on the real line always satisfy a three-term
recurrence relation. The recurrence coefficients determine a tridiagonal
semi-infinite matrix (Jacobi matrix) which uniquely characterizes the
orthogonal polynomials. We investigate new orthogonal polynomials by adding to
the Jacobi matrix new rows and columns, so that the original Jacobi matrix
is shifted downward. The new rows and columns contain new parameters
and the newly obtained orthogonal polynomials thus correspond to an upward
extension of the Jacobi matrix. We give an explicit expression of the new
orthogonal polynomials in terms of the original orthogonal polynomials, their
associated polynomials and the new parameters, and we give a fourth order
differential equation for these new polynomials when the original orthogonal
polynomials are classical. Furthermore we show how the orthogonalizing measure
for these new orthogonal polynomials can be obtained and work out the details
for a one-parameter family of Jacobi polynomials for which the associated
polynomials are again Jacobi polynomials
Bivariate second--order linear partial differential equations and orthogonal polynomial solutions
In this paper we construct the main algebraic and differential properties and
the weight functions of orthogonal polynomial solutions of bivariate
second--order linear partial differential equations, which are admissible
potentially self--adjoint and of hypergeometric type. General formulae for all
these properties are obtained explicitly in terms of the polynomial
coefficients of the partial differential equation, using vector matrix
notation. Moreover, Rodrigues representations for the polynomial eigensolutions
and for their partial derivatives of any order are given. Finally, as
illustration, these results are applied to specific Appell and Koornwinder
orthogonal polynomials, solutions of the same partial differential equation.Comment: 27 page
Families of quasi-exactly solvable extensions of the quantum oscillator in curved spaces
We introduce two new families of quasi-exactly solvable (QES) extensions of
the oscillator in a -dimensional constant-curvature space. For the first
three members of each family, we obtain closed-form expressions of the energies
and wavefunctions for some allowed values of the potential parameters using the
Bethe ansatz method. We prove that the first member of each family has a hidden
sl(2,) symmetry and is connected with a QES equation of the first
or second type, respectively. One-dimensional results are also derived from the
-dimensional ones with , thereby getting QES extensions of the
Mathews-Lakshmanan nonlinear oscillator.Comment: 30 pages, 8 figures, published versio
A model for Hopfions on the space-time S^3 x R
We construct static and time dependent exact soliton solutions for a theory
of scalar fields taking values on a wide class of two dimensional target
spaces, and defined on the four dimensional space-time S^3 x R. The
construction is based on an ansatz built out of special coordinates on S^3. The
requirement for finite energy introduces boundary conditions that determine an
infinite discrete spectrum of frequencies for the oscillating solutions. For
the case where the target space is the sphere S^2, we obtain static soliton
solutions with non-trivial Hopf topological charges. In addition, such hopfions
can oscillate in time, preserving their topological Hopf charge, with any of
the frequencies belonging to that infinite discrete spectrum.Comment: Enlarged version with the time-dependent solutions explicitly given.
One reference and two eps figures added. 14 pages, late
Quasinormal frequencies and thermodynamic quantities for the Lifshitz black holes
We find the connection between thermodynamic quantities and quasinormal
frequencies in Lifshitz black holes. It is shown that the globally stable
Lifshitz black holes have pure imaginary quasinormal frequencies. We also show
that by employing the Maggiore's method, both the horizon area and the entropy
can be quantized for these black holes.Comment: 21 pages, no figures, version to appear in PR
PT-Invariant Periodic Potentials with a Finite Number of Band Gaps
We obtain the band edge eigenstates and the mid-band states for the complex,
PT-invariant generalized associated Lam\'e potentials V^{PT}(x)=-a(a+1)m
\sn^2(y,m)-b(b+1)m {\sn^2 (y+K(m),m)} -f(f+1)m {\sn^2
(y+K(m)+iK'(m),m)}-g(g+1)m {\sn^2 (y+iK'(m),m)}, where ,
and there are four parameters . This work is a substantial
generalization of previous work with the associated Lam\'e potentials
V(x)=a(a+1)m\sn^2(x,m)+b(b+1)m{\sn^2 (x+K(m),m)} and their corresponding
PT-invariant counterparts , both of which involving
just two parameters . We show that for many integer values of ,
the PT-invariant potentials are periodic problems with a finite
number of band gaps. Further, usingsupersymmetry, we construct several
additional, new, complex, PT-invariant, periodic potentials with a finite
number of band gaps. We also point out the intimate connection between the
above generalized associated Lam\'e potential problem and Heun's differential
equation.Comment: 30 pages, 0 figure
Recurrence relations for connection coefficients between q-orthogonal polynomials of discrete variables in the non-uniform lattice X(s) = q2s
We obtain the structure relations for q-orthogonal polynomials in the exponential lattice q 2s and from that we construct the recurrence relation for the connection coe cients between two families of polynomials belonging to the classical class of discrete q-orthogonal polynomials. An explicit example is also given.Comisión Interministerial de Ciencia y Tecnologí
Heun equation, Teukolsky equation, and type-D metrics
Starting with the whole class of type-D vacuum backgrounds with cosmological
constant we show that the separated Teukolsky equation for zero rest-mass
fields with spin (gravitational waves), (electromagnetic
waves) and (neutrinos) is an Heun equation in disguise.Comment: 27 pages, corrected typo in eq. (1
- …