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

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 $d$-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,$\mathbb{R}$) symmetry and is connected with a QES equation of the first or second type, respectively. One-dimensional results are also derived from the $d$-dimensional ones with $d \ge 2$, thereby getting QES extensions of the Mathews-Lakshmanan nonlinear oscillator.Comment: 30 pages, 8 figures, published versio

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 $y \equiv ix+\beta$, and there are four parameters $a,b,f,g$. 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 $V^{PT}(x)=-V(ix+\beta)$, both of which involving just two parameters $a,b$. We show that for many integer values of $a,b,f,g$, the PT-invariant potentials $V^{PT}(x)$ 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

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

Extension of Nikiforov-Uvarov Method for the Solution of Heun Equation

We report an alternative method to solve second order differential equations which have at most four singular points. This method is developed by changing the degrees of the polynomials in the basic equation of Nikiforov-Uvarov (NU) method. This is called extended NU method for this paper. The eigenvalue solutions of Heun equation and confluent Heun equation are obtained via extended NU method. Some quantum mechanical problems such as Coulomb problem on a 3-sphere, two Coulombically repelling electrons on a sphere and hyperbolic double-well potential are investigated by this method

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 $s=\pm 2$ (gravitational waves), $s=\pm 1$ (electromagnetic waves) and $s=\pm 1/2$ (neutrinos) is an Heun equation in disguise.Comment: 27 pages, corrected typo in eq. (1

Spectra generated by a confined softcore Coulomb potential

Analytic and approximate solutions for the energy eigenvalues generated by a confined softcore Coulomb potentials of the form a/(r+\beta) in d>1 dimensions are constructed. The confinement is effected by linear and harmonic-oscillator potential terms, and also through `hard confinement' by means of an impenetrable spherical box. A byproduct of this work is the construction of polynomial solutions for a number of linear differential equations with polynomial coefficients, along with the necessary and sufficient conditions for the existence of such solutions. Very accurate approximate solutions for the general problem with arbitrary potential parameters are found by use of the asymptotic iteration method.Comment: 17 pages, 2 figure

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

Analytic structure of radiation boundary kernels for blackhole perturbations

Exact outer boundary conditions for gravitational perturbations of the Schwarzschild metric feature integral convolution between a time-domain boundary kernel and each radiative mode of the perturbation. For both axial (Regge-Wheeler) and polar (Zerilli) perturbations, we study the Laplace transform of such kernels as an analytic function of (dimensionless) Laplace frequency. We present numerical evidence indicating that each such frequency-domain boundary kernel admits a "sum-of-poles" representation. Our work has been inspired by Alpert, Greengard, and Hagstrom's analysis of nonreflecting boundary conditions for the ordinary scalar wave equation.Comment: revtex4, 14 pages, 12 figures, 3 table

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