384 research outputs found
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
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
Fourth-order differential equations satisfied by the generalized co-recursive of all classical orthogonal polynomials. A study of their distribution of zeros
AbstractThe unique fourth-order differential equation satisfied by the generalized co-recursive of all classical orthogonal polynomials is given for any (but fixed) level of recursivity. Up to now, these differential equations were known only for each classical family separately and also for a specific recursivity level. Moreover, we use this unique fourth-order differential equation in order to study the distribution of zeros of these polynomials via their Newton sum rules (i.e., the sums of powers of their zeros) which are closely related with the moments of such distribution. Both results are obtained with the help of two programs built in Mathematica symbolic language
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
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
Solution of the Dirac equation in the rotating Bertotti-Robinson spacetime
The Dirac equation is solved in the rotating Bertotti-Robinson spacetime. The
set of equations representing the Dirac equation in the Newman-Penrose
formalism is decoupled into an axial and angular part. The axial equation,
which is independent of mass, is solved exactly in terms of hypergeometric
functions. The angular equation is considered both for massless (neutrino) and
massive spin-(1/2) particles. For the neutrinos, it is shown that the angular
equation admits an exact solution in terms of the confluent Heun equation. In
the existence of mass, the angular equation does not allow an analytical
solution, however, it is expressible as a set of first order differential
equations apt for numerical study.Comment: 17 pages, no figure. Appeared in JMP (May, 2008
Complex Periodic Potentials with a Finite Number of Band Gaps
We obtain several new results for the complex generalized associated Lame
potential V(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 y = x-K(m)/2-iK'(m)/2,
sn(y,m) is a Jacobi elliptic function with modulus parameter m, and there are
four real parameters a,b,f,g. First, we derive two new duality relations which,
when coupled with a previously obtained duality relation, permit us to relate
the band edge eigenstates of the 24 potentials obtained by permutations of the
four parameters a,b,f,g. Second, we pose and answer the question: how many
independent potentials are there with a finite number "a" of band gaps when
a,b,f,g are integers? For these potentials, we clarify the nature of the band
edge eigenfunctions. We also obtain several analytic results when at least one
of the four parameters is a half-integer. As a by-product, we also obtain new
solutions of Heun's differential equation.Comment: 33 pages, 0 figure
Angle Dependence of Landau Level Spectrum in Twisted Bilayer Graphene
In the context of the low energy effective theory, the exact Landau level
spectrum of quasiparticles in twisted bilayer graphene with small twist angle
is analytically obtained by spheroidal eigenvalues. We analyze the dependence
of the Landau levels on the twist angle to find the points, where the two-fold
degeneracy for twist angles is lifted in the nonzero modes and below/above
which massive/massless fermion pictures become valid. In the perpendicular
magnetic field of 10\,T, the degeneracy is removed at %angles around 3 degrees for a few low levels, specifically,
for the first pair of nonzero levels and
for the next pair. Massive quasiparticle
appears at in 10\,T, %angles less
than 1.17 degrees. which match perfectly with the recent experimental results.
Since our analysis is applicable to the cases of arbitrary constant magnetic
fields, we make predictions for the same experiment performed in arbitrary
constant magnetic fields, e.g., for B=40\,T we get and the sequence of angles for the pairs of nonzero energy levels. The symmetry restoration
mechanism behind the massive/massless transition is conjectured to be a
tunneling (instanton) in momentum space.Comment: 8 pages, 7 figures, version to appear in PR
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