Volume 48, Issue 4, April 2007
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


On multidimensional inverse scattering for Stark Hamiltonians
View Description Hide DescriptionBased on the EnssWeder [“The geometrical approach to multidimensional inverse scattering,”J. Math. Phys.36, 3902–3921 (1995)] timedependent method, we study one of multidimensional inverse scattering problems for Stark Hamiltonians. We first show that when the space dimension is greater than or equal to 2, the high velocity limit of the scattering operator determines uniquely the potential such as with which is short range under the Stark effect. This is an improvement of previous results obtained by Nicoleau [“Inverse scattering for Stark Hamiltonians with shortrange potentials,” Asymptotic Anal.35, 349–359 (2003)] and Weder [“Multidimensional inverse scattering in an electric field,” J. Funct. Anal.139, 441–465 (1996)]. Moreover, we prove that for a given longrange part of the potential under the Stark effect, the high velocity limit of the Dollardtype modified scattering operator determines uniquely the shortrange part of the potential.

Finite volume approximation of the Anderson model
View Description Hide DescriptionIn the Anderson model on , we consider a sequence of its finite volume approximation and construct a set of sequences composed of the eigenvalues and eigenfunctions of in the localized region which converge to those of simultaneously. For its proof, Minami’s estimate turns out to be important. This result implies that, in the localized region, each eigenfunction behaves almost independently around their centers of localization.

Propagator for finite range potentials: The case of reflection
View Description Hide DescriptionFollowing a previous study on the transmission propagator for a finite range potential, the problem of reflection is considered. It is found that the Laplace transform of the reflection propagator can be expressed in terms of the usual Fredholm determinant and of a new similar determinant, containing the peculiar characteristics of reflection. As an example, an array of delta potentials is considered. Moreover, a possible application to the calculation of quantum traversal time is shown.

Rapidly rotating BoseEinstein condensates in strongly anharmonic traps
View Description Hide DescriptionWe study a rotating BoseEinstein condensate in a strongly anharmonic trap (flat trap with a finite radius) in the framework of twodimensional GrossPitaevskii theory. We write the coupling constant for the interactions between the gas atoms as and we are interested in the limit (ThomasFermi limit) with the angular velocity depending on . We derive rigorously the leading asymptotics of the ground state energy and the density profile when tends to infinity as a power of . If a “hole” (i.e., a region where the density becomes exponentially small as ) develops for above a certain critical value. If the hole essentially exhausts the container and a “giant vortex” develops with the density concentrated in a thin layer at the boundary. While we do not analyze the detailed vortex structure we prove that rotational symmetry is broken in the ground state for .
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Generalized Weierstrass representations of surfaces with the constant Gauss curvature in pseudoRiemannian threedimensional space forms
View Description Hide DescriptionWe know that there exists the correspondence between one of the nonumbilical Riemannian surface with constant Gauss curvature in , the nonumbilical spacelike or timelike surfaces with constant Gauss curvature in , and a solution of one of the following differential equations: the sineGordon, sinhGordon, sinelaplace, sinhlaplace, and coshGordon equations. In this paper, we consider the initial value problems of these equations and give the generalized Weierstrass representations of these surfaces that depend only on the initial values of these equations.
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 GENERAL RELATIVITY AND GRAVITATION


Tikekar superdense stars in electric fields
View Description Hide DescriptionWe present exact solutions to the EinsteinMaxwell system of equations with a specified form of the electric field intensity by assuming that the hypersurface are spheroidal. The solution of the EinsteinMaxwell system is reduced to a recurrence relation with variable rational coefficients which can be solved in general using mathematical induction. New classes of solutions of linearly independent functions are obtained by restricting the spheroidal parameter and the electric field intensity parameter . Consequently, it is possible to find exact solutions in terms of elementary functions, namely, polynomials and algebraic functions. Our result contains models found previously including the superdense Tikekar neutron starmodel [J. Math. Phys.31, 2454 (1990)] when and . Our class of charged spheroidal models generalize the uncharged isotropic Maharaj and Leach solutions [J. Math. Phys.37, 430 (1996)]. In particular, we find an explicit relationship directly relating the spheroidal parameter to the electromagnetic field.

Heun equation, Teukolsky equation, and typeD metrics
View Description Hide DescriptionStarting with the whole class of typeD vacuum backgrounds with cosmological constant we show that the separated Teukolsky equation for zero restmass fields with spins (gravitational waves), (electromagnetic waves), and (neutrinos) is a Heun equation in disguise.
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 DYNAMICAL SYSTEMS


Removal of resonances by rotation in linearly degenerate twodimensional oscillator systems
View Description Hide DescriptionA system of two nonlinearly interacting, resonant harmonic oscillators is investigated, seeking transformation to approximate actionangle variables in the vicinity of the equilibrium via the canonical perturbation theory. A variety of polynomial perturbations dependent on parameters is considered. The freedom of choice of the zeroorder approximation characteristic of a linearly degenerate (resonant) system is used to cancel lowerorder resonant terms in the canonical perturbation series. It is found that the cancellation of the resonant terms is only possible for particular values of parameters of the interaction term. These special sets of parameters include all the cases with the Panlevé property.

SasaSatsuma (complex modified Korteweg–de Vries II) and the complex sineGordon II equation revisited: Recursion operators, nonlocal symmetries, and more
View Description Hide DescriptionWe present a new symplectic structure and a hereditary recursion operator for the SasaSatsuma equation which is widely used in nonlinear optics. Using an integrodifferential substitution relating this equation to a thirdorder symmetry flow of the complex sineGordon II equation enabled us to find a hereditary recursion operator and higher Hamiltonian structures for the latter equation. We also show that both the SasaSatsuma equation and the thirdorder symmetry flow for the complex sineGordon II equation are biHamiltonian systems, and we construct several hierarchies of local and nonlocal symmetries for these systems.

Spontaneous breaking of classical symmetry
View Description Hide DescriptionThe classical trajectories of the family of complex symmetric Hamiltonians form closed orbits. All such complex orbits that have been studied in the past are symmetric (leftright symmetric). The periods of these orbits exhibit an unusual dependence on the parameter . There are regions in of smooth behavior interspersed with regions of rapid variation. It is demonstrated that the onset of rapid variation is associated with strange new kinds of classical trajectories that have never been seen previously. These rare kinds of trajectories are not symmetric and occur only for special rational values of .

Equivalence of energy methods in stability theory
View Description Hide DescriptionWe will prove the equivalence of three methods, the so called energy methods, in order to establish the stability of an equilibrium point for a dynamical system. We will illustrate by examples that this result simplifies enormously the amount of computations especially when the stability cannot be decided with one of the three methods.
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 STATISTICAL PHYSICS


Lyapunov exponents for unitary Anderson models
View Description Hide DescriptionWe study a unitary version of the onedimensional Anderson model, given by a five diagonal deterministic unitary operator multiplicatively perturbed by a random phase matrix. We fully characterize positivity and vanishing of the Lyapunov exponent for this model throughout the spectrum and for arbitrary distributions of the random phases. This includes Bernoulli distributions, where in certain cases a finite number of critical spectral values, with vanishing Lyapunov exponent, exist. We establish similar results for a unitary version of the random dimer model.

Domain wall and periodic solutions of coupled asymmetric double well models
View Description Hide DescriptionCoupled asymmetric double well onedimensional potentials arise in the context of first order phase transitions both in condensed matter physics and field theory. Here we provide an exhaustive set of exact periodic solutions of such a coupled asymmetric model in terms of elliptic functions (domain wall arrays) and obtain single domain wallsolutions in specific limits. We also calculate the energy and interaction between solitons for various solutions. Both topological (kinklike at ) and nontopological (pulselike for ) domain wallsolutions are obtained. We relate some of these solutions to domain walls in hydrogen bonded materials and also in the field theory context. As a byproduct, we also obtain a new one parameter family of kink solutions of the uncoupled asymmetric double well model.

How to determine the law of the solution to a stochastic partial differential equation driven by a Lévy spacetime noise?
View Description Hide DescriptionWe consider a stochastic partial differential equation on a lattice , where is a spacetime Lévy noise. A perturbative (in the sense of formal power series) strong solution is given by a tree expansion, whereas the correlation functions of the solution are given by a perturbative expansion with coefficients that are represented as sums over a certain class of graphs, called ParisiWu graphs. The perturbative expansion of the truncated (connected) correlation functions is obtained via a linked cluster theorem as sums over connected graphs only. The moments of the stationary solution can be calculated as well. In all these solutions the cumulants of the single site distribution of the noise enter as multiplicative constants. To determine them, e.g., by comparison with an empirical correlation function, one can fit these constants (e.g., by the methods of least squares) and thereby one (approximately) determines laws of the solution and the driving noise.
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 METHODS OF MATHEMATICAL PHYSICS


Reconstruction of radial Dirac operators
View Description Hide DescriptionWe study the inverse spectral problem of reconstructing the potential of radial Dirac operators acting in the unit ball of . For each onedimensional partial Dirac operator corresponding to a nonzero angular momentum, we give a complete description of the spectral data (eigenvalues and suitably defined norming constants), prove existence and uniqueness of solutions to the inverse problem, and present the reconstruction algorithm.

Generalized fractional Schrödinger equation with spacetime fractional derivatives
View Description Hide DescriptionIn this paper the generalized fractional Schrödinger equation with space and time fractional derivatives is constructed. The equation is solved for free particle and for a square potential well by the method of integral transforms,Fourier transform and Laplace transform, and the solution can be expressed in terms of MittagLeffler function. The Green function for free particle is also presented in this paper. Finally, we discuss the relationship between the cases of the generalized fractional Schrödinger equation and the ones in standard quantum.

Informationtheoretic measures of hyperspherical harmonics
View Description Hide DescriptionThe multidimensional spreading of the hyperspherical harmonics can be measured in a different and complementary manner by means of the following informationtheoretic quantities: the Fisher information, the average density or firstorder entropic moment, and the Shannon entropy. They give measures of the volume anisotropy of the eigenfunctions of any central potential in the hyperspace. Contrary to the Fisher information, which is a local measure because of its gradientfunctional form, the other two quantities have a global character because they are powerlike (average density) and logarithmic (Shannon’s entropy) functionals of the hyperspherical harmonics. In this paper we obtain the explicit expression of the first two measures and a lower bound to the Shannon entropy in terms of the labeling indices of the hyperspherical harmonics.

Classification of generalized quantum statistics associated with the exceptional Lie (super)algebras
View Description Hide DescriptionGeneralized quantum statistics (GQS) associated with a Lie algebra or Lie superalgebra extends the notion of paraBose or paraFermi statistics. Such GQS have been classified for all classical simple Lie algebras and basic classical Lie superalgebras. In the current paper we finalize this classification for all exceptional Lie algebras and superalgebras. Since the definition of GQS is closely related to a certain grading of the Lie (super)algebra , our classification reproduces some known gradings of exceptional Lie algebras. For exceptional Lie superalgebras such a classification of gradings has not been given before.

Invariant analytic orthonormalization procedure with an application to coherent states
View Description Hide DescriptionWe discuss a general strategy which produces an orthonormal set of vectors, stable under the action of a given set of unitary operators , , starting from a fixed normalized vector in and from a set of unitary operators. We discuss several examples of this procedure and, in particular, we show how a set of coherentlike vectors can be produced and in which condition over the lattice spacing this can be done.

Closedform summation of the Dowker and related sums
View Description Hide DescriptionFinite sums of powers of cosecants appear in a wide range of physical problems. We, through a unified approach which uses contour integrals and residues, establish the summation formulas for two general families of such sums. One of them is the family which was first studied and summed in closed form by Dowker [Phys. Rev. D36, 3095 (1987)], while the other is related to it and has not been studied before. Our summation formulas of the Dowker sums involve only the Stirling numbers of the first kind and the (ordinary) Bernoullipolynomials and numbers, unlike the earlier summation formulas in which either the higherorder Bernoulli numbers and polynomials or the multiple sums involving the Bernoulli numbers and their products, were used. A great deal of other (known or presumably new) closedform summations follows as straightforward corollaries to these formulas. Among them are two special cases of the celebrated Verlinde’s formula and numerous sums encountered in various physical problems by McCoy and Orrick [J. Stat. Phys.83, 839 (1996)], Gervois and Mehta [J. Math. Phys.36, 5098 (1995)], and Henkel and Lacki [Phys. Lett. A138, 105 (1989)].
