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Apr 2007

Volume 48, Issue 4, Articles (04xxxx)

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On multidimensional inverse scattering for Stark Hamiltonians

Tadayoshi Adachi and Katsuhiro Maehara

J. Math. Phys. 48, 042101 (2007); http://dx.doi.org/10.1063/1.2713077 (12 pages) | Cited 2 times

Online Publication Date: 6 April 2007

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Based on the Enss-Weder [“The geometrical approach to multidimensional inverse scattering,” J. Math. Phys. 36, 3902–3921 (1995)] time-dependent 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 ∣x∣−γ with γ>1/2 which is short range under the Stark effect. This is an improvement of previous results obtained by Nicoleau [“Inverse scattering for Stark Hamiltonians with short-range 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 long-range part of the potential under the Stark effect, the high velocity limit of the Dollard-type modified scattering operator determines uniquely the short-range part of the potential.
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03.65.Nk Scattering theory

Finite volume approximation of the Anderson model

Fumihiko Nakano

J. Math. Phys. 48, 042102 (2007); http://dx.doi.org/10.1063/1.2716970 (6 pages)

Online Publication Date: 13 April 2007

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In the Anderson model on Zd, we consider a sequence of its finite volume approximation {Hk}k and construct a set of sequences composed of the eigenvalues and eigenfunctions of {Hk} in the localized region I which converge to those of H 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.
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02.60.-x Numerical approximation and analysis
02.10.Ud Linear algebra

Propagator for finite range potentials: The case of reflection

Ilaria Cacciari and Paolo Moretti

J. Math. Phys. 48, 042103 (2007); http://dx.doi.org/10.1063/1.2728518 (7 pages) | Cited 1 time

Online Publication Date: 26 April 2007

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Following 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 Δ and of a new similar determinant Γ, 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.
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03.65.Ge Solutions of wave equations: bound states
02.30.Uu Integral transforms

Rapidly rotating Bose-Einstein condensates in strongly anharmonic traps

M. Correggi, T. Rindler-Daller, and J. Yngvason

J. Math. Phys. 48, 042104 (2007); http://dx.doi.org/10.1063/1.2712421 (30 pages) | Cited 14 times

Online Publication Date: 30 April 2007

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We study a rotating Bose-Einstein condensate in a strongly anharmonic trap (flat trap with a finite radius) in the framework of two-dimensional Gross-Pitaevskii theory. We write the coupling constant for the interactions between the gas atoms as 1/ε2 and we are interested in the limit ε→0 (Thomas-Fermi limit) with the angular velocity Ω depending on ε. We derive rigorously the leading asymptotics of the ground state energy and the density profile when Ω tends to infinity as a power of 1/ε. If Ω(ε) = Ω0/ε a “hole” (i.e., a region where the density becomes exponentially small as 1/ε→∞) develops for Ω0 above a certain critical value. If Ω(ε)⪢1/ε 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 const∣log ε∣<Ω(ε)≲const/ε.
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05.30.Jp Boson systems
02.30.-f Function theory, analysis
03.75.Hh Static properties of condensates; thermodynamical, statistical, and structural properties
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Generalized Weierstrass representations of surfaces with the constant Gauss curvature in pseudo-Riemannian three-dimensional space forms

Qiaoling Xia

J. Math. Phys. 48, 042301 (2007); http://dx.doi.org/10.1063/1.2714002 (18 pages)

Online Publication Date: 10 April 2007

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We know that there exists the correspondence between one of the nonumbilical Riemannian surface with constant Gauss curvature K in math3(math)(K ≠ math), the nonumbilical spacelike or timelike surfaces with constant Gauss curvature K in math13(math)(K ≠ math), and a solution of one of the following differential equations: the sine-Gordon, sinh-Gordon, sine-laplace, sinh-laplace, and cosh-Gordon 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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04.20.Gz Spacetime topology, causal structure, spinor structure
02.30.-f Function theory, analysis
02.60.Lj Ordinary and partial differential equations; boundary value problems
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Tikekar superdense stars in electric fields

K. Komathiraj and S. D. Maharaj

J. Math. Phys. 48, 042501 (2007); http://dx.doi.org/10.1063/1.2716204 (12 pages) | Cited 9 times

Online Publication Date: 12 April 2007

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We present exact solutions to the Einstein-Maxwell system of equations with a specified form of the electric field intensity by assuming that the hypersurface {t = constant} are spheroidal. The solution of the Einstein-Maxwell 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 K and the electric field intensity parameter α. 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 star model [ J. Math. Phys. 31, 2454 (1990) ] when K = −7 and α = 0. 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 K to the electromagnetic field.
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97.60.Jd Neutron stars
97.10.Cv Stellar structure, interiors, evolution, nucleosynthesis, ages
97.10.Ld Magnetic and electric fields; polarization of starlight

Heun equation, Teukolsky equation, and type-D metrics

D. Batic and H. Schmid

J. Math. Phys. 48, 042502 (2007); http://dx.doi.org/10.1063/1.2720277 (23 pages) | Cited 6 times

Online Publication Date: 25 April 2007

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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 spins s = ±2 (gravitational waves), s = ±1 (electromagnetic waves), and s = ±1/2 (neutrinos) is a Heun equation in disguise.
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98.80.Jk Mathematical and relativistic aspects of cosmology
98.80.Es Observational cosmology (including Hubble constant, distance scale, cosmological constant, early Universe, etc)
04.62.+v Quantum fields in curved spacetime
04.30.-w Gravitational waves
02.30.Hq Ordinary differential equations
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Removal of resonances by rotation in linearly degenerate two-dimensional oscillator systems

Michael Khasin and Lazar Friedland

J. Math. Phys. 48, 042701 (2007); http://dx.doi.org/10.1063/1.2719145 (11 pages) | Cited 2 times

Online Publication Date: 11 April 2007

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A system of two nonlinearly interacting, resonant harmonic oscillators is investigated, seeking transformation to approximate action-angle 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 zero-order approximation characteristic of a linearly degenerate (resonant) system is used to cancel lower-order 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.
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45.05.+x General theory of classical mechanics of discrete systems
02.10.De Algebraic structures and number theory

Sasa-Satsuma (complex modified Korteweg–de Vries II) and the complex sine-Gordon II equation revisited: Recursion operators, nonlocal symmetries, and more

Artur Sergyeyev and Dmitry Demskoi

J. Math. Phys. 48, 042702 (2007); http://dx.doi.org/10.1063/1.2710552 (11 pages) | Cited 5 times

Online Publication Date: 16 April 2007

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We present a new symplectic structure and a hereditary recursion operator for the Sasa-Satsuma equation which is widely used in nonlinear optics. Using an integrodifferential substitution relating this equation to a third-order symmetry flow of the complex sine-Gordon II equation enabled us to find a hereditary recursion operator and higher Hamiltonian structures for the latter equation. We also show that both the Sasa-Satsuma equation and the third-order symmetry flow for the complex sine-Gordon II equation are bi-Hamiltonian systems, and we construct several hierarchies of local and nonlocal symmetries for these systems.
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42.65.-k Nonlinear optics
02.30.-f Function theory, analysis
02.30.Jr Partial differential equations

Spontaneous breaking of classical PT symmetry

Carl M. Bender and Daniel W. Darg

J. Math. Phys. 48, 042703 (2007); http://dx.doi.org/10.1063/1.2720279 (14 pages) | Cited 15 times

Online Publication Date: 25 April 2007

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The classical trajectories of the family of complex PT-symmetric Hamiltonians H = p2+x2(ix)ϵ (ϵ ≥ 0) form closed orbits. All such complex orbits that have been studied in the past are PT symmetric (left-right symmetric). The periods of these orbits exhibit an unusual dependence on the parameter ϵ. There are regions in ϵ 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 PT symmetric and occur only for special rational values of ϵ.
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11.30.Er Charge conjugation, parity, time reversal, and other discrete symmetries
11.30.Qc Spontaneous and radiative symmetry breaking
02.30.Hq Ordinary differential equations

Equivalence of energy methods in stability theory

P. Birtea and M. Puta

J. Math. Phys. 48, 042704 (2007); http://dx.doi.org/10.1063/1.2716201 (9 pages) | Cited 3 times

Online Publication Date: 27 April 2007

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We 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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05.45.-a Nonlinear dynamics and chaos
02.30.Sa Functional analysis
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Lyapunov exponents for unitary Anderson models

Eman Hamza and Günter Stolz

J. Math. Phys. 48, 043301 (2007); http://dx.doi.org/10.1063/1.2713996 (16 pages) | Cited 1 time

Online Publication Date: 20 April 2007

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We study a unitary version of the one-dimensional 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.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.10.Yn Matrix theory

Domain wall and periodic solutions of coupled asymmetric double well models

Avinash Khare and Avadh Saxena

J. Math. Phys. 48, 043302 (2007); http://dx.doi.org/10.1063/1.2716202 (19 pages) | Cited 3 times

Online Publication Date: 23 April 2007

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Coupled asymmetric double well (aϕ2−bϕ3+cϕ4) one-dimensional 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 wall solutions in specific limits. We also calculate the energy and interaction between solitons for various solutions. Both topological (kinklike at T = Tc) and nontopological (pulselike for T ≠ Tc) domain wall solutions are obtained. We relate some of these solutions to domain walls in hydrogen bonded materials and also in the field theory context. As a by-product, we also obtain a new one parameter family of kink solutions of the uncoupled asymmetric double well model.
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64.60.-i General studies of phase transitions
02.30.Sa Functional analysis
02.40.Pc General topology

How to determine the law of the solution to a stochastic partial differential equation driven by a Lévy space-time noise?

Hanno Gottschalk and Boubaker Smii

J. Math. Phys. 48, 043303 (2007); http://dx.doi.org/10.1063/1.2712916 (22 pages)

Online Publication Date: 30 April 2007

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We consider a stochastic partial differential equation on a lattice ∂tX = (Δ−m2)X−λXp+η, where η is a space-time 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 Parisi-Wu 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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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.10.Ox Combinatorics; graph theory
02.30.Jr Partial differential equations
02.50.Ey Stochastic processes
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Reconstruction of radial Dirac operators

S. Albeverio, R. Hryniv, and Ya. Mykytyuk

J. Math. Phys. 48, 043501 (2007); http://dx.doi.org/10.1063/1.2709847 (14 pages) | Cited 1 time

Online Publication Date: 3 April 2007

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We study the inverse spectral problem of reconstructing the potential of radial Dirac operators acting in the unit ball of math3. For each one-dimensional 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.
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03.65.Pm Relativistic wave equations
02.30.Zz Inverse problems
03.65.Fd Algebraic methods

Generalized fractional Schrödinger equation with space-time fractional derivatives

Shaowei Wang and Mingyu Xu

J. Math. Phys. 48, 043502 (2007); http://dx.doi.org/10.1063/1.2716203 (10 pages) | Cited 17 times

Online Publication Date: 4 April 2007

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In 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 Mittag-Leffler 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.
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03.65.Ge Solutions of wave equations: bound states
03.65.Db Functional analytical methods
02.30.Uu Integral transforms
02.30.Nw Fourier analysis

Information-theoretic measures of hyperspherical harmonics

J. S. Dehesa, S. López-Rosa, and R. J. Yáñez

J. Math. Phys. 48, 043503 (2007); http://dx.doi.org/10.1063/1.2712913 (10 pages) | Cited 7 times

Online Publication Date: 5 April 2007

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The multidimensional spreading of the hyperspherical harmonics can be measured in a different and complementary manner by means of the following information-theoretic quantities: the Fisher information, the average density or first-order 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 gradient-functional 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.
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03.67.-a Quantum information
02.10.Ud Linear algebra
03.65.Fd Algebraic methods
02.30.Sa Functional analysis

Classification of generalized quantum statistics associated with the exceptional Lie (super)algebras

N. I. Stoilova and J. Van der Jeugt

J. Math. Phys. 48, 043504 (2007); http://dx.doi.org/10.1063/1.2712914 (18 pages) | Cited 1 time

Online Publication Date: 9 April 2007

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Generalized quantum statistics (GQS) associated with a Lie algebra or Lie superalgebra extends the notion of para-Bose or para-Fermi 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 math grading of the Lie (super)algebra G, our classification reproduces some known math gradings of exceptional Lie algebras. For exceptional Lie superalgebras such a classification of math gradings has not been given before.
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05.30.Fk Fermion systems and electron gas
05.30.Jp Boson systems
02.10.Ud Linear algebra
02.20.Sv Lie algebras of Lie groups
03.65.Fd Algebraic methods

Invariant analytic orthonormalization procedure with an application to coherent states

F. Bagarello and S. Triolo

J. Math. Phys. 48, 043505 (2007); http://dx.doi.org/10.1063/1.2711371 (16 pages) | Cited 2 times

Online Publication Date: 17 April 2007

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We discuss a general strategy which produces an orthonormal set of vectors, stable under the action of a given set of unitary operators Aj, j = 1,2,…,n, starting from a fixed normalized vector in H 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.
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02.10.Ab Logic and set theory
02.10.Ud Linear algebra

Nonunitary similarity transformation of conservative to dissipative evolutions: Intertwining without time operator

Fernando Gómez

J. Math. Phys. 48, 043506 (2007); http://dx.doi.org/10.1063/1.2709634 (19 pages) | Cited 1 time

Online Publication Date: 18 April 2007

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Reversible evolutions are usually expressed in terms of unitary groups on separable Hilbert spaces, whereas irreversible ones are described by contraction semigroups. In the theory of nonunitary similarity transformations intertwining unitary groups and contraction semigroups, proposed initially in the context of statistical mechanics as part of an exact theory of irreversibility, the unitary groups with such intertwining property have been qualified by the existence of an internal time operator. This work tackles the question of existence of internal time operators for unitary groups with the intertwining property. Equivalent conditions to the existence of internal time operators for such unitary groups are given on the basis of the Sz.-Nagy–Foiaş [Harmonic Analysis of Operators on Hilbert Spaces (North-Holland, Amsterdam, 1970) ] dilation theory and the theory of shift invariant subspaces. These conditions permit us to solve the inverse intertwining problem in the negative: there are unitary groups with the intertwining property which do not admit internal time operator. A representative family of such unitary groups is given.
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05.20.-y Classical statistical mechanics
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
05.70.Ln Nonequilibrium and irreversible thermodynamics
02.20.-a Group theory
02.50.Ga Markov processes

Closed-form summation of the Dowker and related sums

Djurdje Cvijović and H. M. Srivastava

J. Math. Phys. 48, 043507 (2007); http://dx.doi.org/10.1063/1.2712895 (12 pages) | Cited 1 time

Online Publication Date: 19 April 2007

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Finite 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. D 36, 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) Bernoulli polynomials and numbers, unlike the earlier summation formulas in which either the higher-order 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) closed-form 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. A 138, 105 (1989)] .
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02.10.De Algebraic structures and number theory
02.30.Lt Sequences, series, and summability
02.30.Rz Integral equations

Classification of modules of the intermediate series over Ramond N = 2 superconformal algebras

Jiayuan Fu, Qifen Jiang, and Yucai Su

J. Math. Phys. 48, 043508 (2007); http://dx.doi.org/10.1063/1.2713998 (15 pages) | Cited 2 times

Online Publication Date: 24 April 2007

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In this paper, we first discuss the structure of the Ramond N = 2 superconformal algebras. Then we classify the modules of the intermediate series over Ramond N = 2 superconformal algebra.
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11.30.Pb Supersymmetry
11.25.Hf Conformal field theory, algebraic structures
02.10.-v Logic, set theory, and algebra

Scattering rule in soliton cellular automaton associated with crystal base of Uq(D4(3))

Daisuke Yamada

J. Math. Phys. 48, 043509 (2007); http://dx.doi.org/10.1063/1.2721347 (28 pages) | Cited 2 times

Online Publication Date: 26 April 2007

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In terms of the crystal base of a quantum affine algebra Uq(math), we study a soliton cellular automaton (SCA) associated with the exceptional affine Lie algebra math = D4(3). The solitons therein are labeled by the crystals of quantum affine algebra Uq(A1(1)). The scatteing rule is identified with the combinatorial R matrix for Uq(A1(1)) crystals. Remarkably, the phase shifts in our SCA are given by three times of those in the well-known box-ball system.
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61.50.Ah Theory of crystal structure, crystal symmetry; calculations and modeling
02.10.Ud Linear algebra
02.10.Yn Matrix theory

On a q-analog of a Sahi result

Olga Bershtein

J. Math. Phys. 48, 043510 (2007); http://dx.doi.org/10.1063/1.2716200 (8 pages)

Online Publication Date: 27 April 2007

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We obtain a q-analog of a well-known Sahi result on the joint spectrum of S(GLn×GLn)-invariant differential operators with polynomial coefficients on the vector space of complex n×n matrices.
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03.65.Fd Algebraic methods
02.20.Uw Quantum groups
02.10.Yn Matrix theory
02.30.Hq Ordinary differential equations
02.30.Tb Operator theory

Orders and dimensions for sl(2) or sl(3) module categories and boundary conformal field theories on a torus

R. Coquereaux and G. Schieber

J. Math. Phys. 48, 043511 (2007); http://dx.doi.org/10.1063/1.2714000 (17 pages) | Cited 5 times

Online Publication Date: 30 April 2007

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After giving a short description, in terms of action of categories, of some of the structures associated with sl(2) and sl(3) boundary conformal field theories on a torus, we provide tables of dimensions describing the semisimple and cosemisimple blocks of the corresponding weak bialgebras (quantum groupoids), tables of quantum dimensions and orders, and tables describing induction-restriction. For reasons of size, the sl(3) tables of induction are only given for theories with self-fusion (existence of a monoidal structure).
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11.25.Hf Conformal field theory, algebraic structures
03.65.-w Quantum mechanics
02.10.-v Logic, set theory, and algebra
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