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Aug 2008

Volume 49, Issue 8, Articles (08xxxx)

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Minimal length uncertainty relations and new shape invariant models

Donald Spector

J. Math. Phys. 49, 082101 (2008); http://dx.doi.org/10.1063/1.2955795 (8 pages) | Cited 2 times

Online Publication Date: 7 August 2008

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This paper identifies a new class of shape invariant models. These models are based on extensions of conventional quantum mechanics that satisfy a string-motivated minimal length uncertainty relation. An important feature of our construction is the pairing of operators that are not adjoints of each other. The results in this paper thus show the broader applicability of shape invariance to exactly solvable systems.
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03.65.Ge Solutions of wave equations: bound states
03.65.Fd Algebraic methods

Extending the class of solvable potentials. I. The infinite potential well with a sinusoidal bottom

A. D. Alhaidari and H. Bahlouli

J. Math. Phys. 49, 082102 (2008); http://dx.doi.org/10.1063/1.2963967 (13 pages) | Cited 6 times

Online Publication Date: 8 August 2008

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This is the first in a series of papers where we succeed in enlarging the class of exactly solvable potentials in one and three dimensions by obtaining solutions for new relativistic and nonrelativistic problems. This is accomplished by constructing a matrix representation of the wave operator in a complete square integrable basis that makes it tridiagonal. Expanding the wave function in this basis makes the wave equation equivalent to a three-term recursion relation for the expansion coefficients. Consequently, finding solutions of the recursion relation is equivalent to solving the original problem. Doing so results in a larger class of solvable potentials. The usual diagonal representation constraint results in a reduction from the larger class to the conventional class of solvable potentials, giving the well-known energy spectra and the corresponding wave functions. Moreover, some of the new solvable problems show evidence of a Klauder-like phenomenon. In the present work, we give an exact solution for the infinite potential well with a bottom that has a sinusoidal shape.
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03.65.Ge Solutions of wave equations: bound states
03.65.Fd Algebraic methods

On the epistemic view of quantum states

Michael Skotiniotis, Aidan Roy, and Barry C. Sanders

J. Math. Phys. 49, 082103 (2008); http://dx.doi.org/10.1063/1.2966133 (13 pages)

Online Publication Date: 18 August 2008

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We investigate the strengths and limitations of the Spekkens toy model, which is a local hidden variable model that replicates many important properties of quantum dynamics. First, we present a set of five axioms that fully encapsulate Spekkens’ toy model. We then test whether these axioms can be extended to capture more quantum phenomena by allowing operations on epistemic as well as ontic states. We discover that the resulting group of operations is isomorphic to the projective extended Clifford group for two qubits. This larger group of operations results in a physically unreasonable model; consequently, we claim that a relaxed definition of valid operations in Spekkens’ toy model cannot produce an equivalence with the Clifford group for two qubits. However, the new operations do serve as tests for correlation in a two toy bit model, analogous to the well known Horodecki criterion for the separability of quantum states.
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03.65.Fd Algebraic methods
03.65.Ta Foundations of quantum mechanics; measurement theory
03.67.-a Quantum information

Fisher information of special functions and second-order differential equations

R. J. Yáñez, P. Sánchez-Moreno, A. Zarzo, and J. S. Dehesa

J. Math. Phys. 49, 082104 (2008); http://dx.doi.org/10.1063/1.2968341 (16 pages) | Cited 6 times

Online Publication Date: 18 August 2008

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We investigate a basic question of analytic information theory, namely, the evaluation of the Fisher information and the relative Fisher information with respect to a non-negative function, for the probability distributions obtained by squaring the special functions of mathematical physics which are solutions of second-order differential equations. We obtain explicit expressions for these information-theoretic properties via the expectation values of the coefficients of the differential equation. We illustrate our approach for various nonrelativistic D-dimensional wavefunctions and some special functions of physicomathematical interest. Emphasis is made in the Nikiforov–Uvarov hypergeometric-type functions, which include and generalize the Hermite functions and the Gauss and Kummer hypergeometric functions, among others.
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03.65.-w Quantum mechanics
02.30.Hq Ordinary differential equations
03.67.-a Quantum information

Geometric phase for non-Hermitian Hamiltonians and its holonomy interpretation

Hossein Mehri-Dehnavi and Ali Mostafazadeh

J. Math. Phys. 49, 082105 (2008); http://dx.doi.org/10.1063/1.2968344 (17 pages) | Cited 12 times

Online Publication Date: 18 August 2008

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For an arbitrary possibly non-Hermitian matrix Hamiltonian H that might involve exceptional points, we construct an appropriate parameter space math and line bundle Ln over math such that the adiabatic geometric phases associated with the eigenstates of the initial Hamiltonian coincide with the holonomies of Ln. We examine the case of 2×2 matrix Hamiltonians in detail and show that, contrary to claims made in some recent publications, geometric phases arising from encircling exceptional points are generally geometrical and not topological in nature.
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03.65.Fd Algebraic methods
02.10.Yn Matrix theory
02.40.-k Geometry, differential geometry, and topology

Wigner oscillators, twisted Hopf algebras, and second quantization

P. G. Castro, B. Chakraborty, and F. Toppan

J. Math. Phys. 49, 082106 (2008); http://dx.doi.org/10.1063/1.2970042 (19 pages) | Cited 10 times

Online Publication Date: 25 August 2008

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By correctly identifying the role of the central extension in the centrally extended Heisenberg algebra h, we show that it is indeed possible to construct a Hopf algebraic structure on the corresponding enveloping algebra U(h) and eventually deform it through the Drinfeld twist. This Hopf algebraic structure and its deformed version UF(h) are shown to be induced from a more “fundamental” Hopf algebra obtained from the Schrödinger field/oscillator algebra and its deformed version provided that the fields/oscillators are regarded as odd elements of a given superalgebra. We also discuss the possible implications in the context of quantum statistics.
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03.65.Fd Algebraic methods
02.10.Ud Linear algebra
05.30.-d Quantum statistical mechanics
03.65.Ge Solutions of wave equations: bound states
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Asymptotic counting of BPS operators in superconformal field theories

James Lucietti and Mukund Rangamani

J. Math. Phys. 49, 082301 (2008); http://dx.doi.org/10.1063/1.2970775 (30 pages) | Cited 2 times

Online Publication Date: 27 August 2008

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We consider some aspects of counting BPS operators which are annihilated by two supercharges in superconformal field theories. For nonzero coupling, the corresponding multivariable partition functions can be written in terms of generating functions for vector partitions or their weighted generalizations. We derive asymptotics for the density of states for a wide class of such multivariable partition functions. We also point out a particular factorization property of the finite N partition functions. Finally, we discuss the concept of a limit curve arising from the large N partition functions, which is related to the notion of a “typical state,” and discuss some implications for the holographic duals.
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11.25.Hf Conformal field theory, algebraic structures
11.30.Pb Supersymmetry
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Rotating black branes in Brans–Dicke–Born–Infeld theory

S. H. Hendi

J. Math. Phys. 49, 082501 (2008); http://dx.doi.org/10.1063/1.2968342 (12 pages) | Cited 6 times

Online Publication Date: 14 August 2008

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In this paper, we present a new class of charged rotating black brane solutions in the higher dimensional Brans–Dicke–Born–Infeld theory and investigate their properties. Solving the field equations directly is a nontrivial task because they include the second derivatives of the scalar field. We remove this difficulty through a conformal transformation. Also, we find that the suitable Lagrangian of Einstein–Born–Infeld–dilaton gravity is not the same as presented by Dehghani et al. [J. Cosmol. Astropart. Phys. 0702, 020 (2007) ]. We show that the given solutions can present black brane, with inner and outer event horizons, an extreme black brane, or a naked singularity provided the parameters of the solutions are chosen suitably. These black brane solutions are neither asymptotically flat nor (anti-)de Sitter. Then we calculate finite Euclidean action, the conserved, and thermodynamic quantities through the use of counterterm method. Finally, we argue that these quantities satisfy the first law of thermodynamics, and the entropy does not follow the area law.
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04.70.-s Physics of black holes
04.20.Fy Canonical formalism, Lagrangians, and variational principles
04.25.-g Approximation methods; equations of motion
05.70.Ce Thermodynamic functions and equations of state
02.40.Pc General topology
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Dispersionful analog of the Whitham hierarchy

Błażej M. Szablikowski and Maciej Błaszak

J. Math. Phys. 49, 082701 (2008); http://dx.doi.org/10.1063/1.2970774 (20 pages) | Cited 1 time

Online Publication Date: 28 August 2008

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The transition from the zero-genus universal Whitham hierarchy to its dispersionful counterpart, making use only of the Lax representations, is presented. This is an alternative approach to that of Takasaki who has recently shown that the dispersionless limit of the charged multicomponent Kadomtsev-Petviashvili (KP) hierarchy is the Whitham hierarchy. The advantage of the presented approach is the construction of finite-field reductions, which are the main concern in this paper. The theory is illustrated by several significant examples.
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05.45.Yv Solitons
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Homogeneous fractional embeddings

Pierre Inizan

J. Math. Phys. 49, 082901 (2008); http://dx.doi.org/10.1063/1.2963497 (14 pages) | Cited 4 times

Online Publication Date: 11 August 2008

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Fractional equations appear in the description of the dynamics of various physical systems. For Lagrangian systems, the embedding theory developed by Cresson [“Fractional embedding of differential operators and Lagrangian systems,” J. Math. Phys. 48, 033504 (2007)] provides a univocal way to obtain such equations, stemming from a least action principle. However, no matter how equations are obtained, the dimension of the fractional derivative differs from the classical one and may induce problems of temporal homogeneity in fractional objects. In this paper, we show that it is necessary to introduce an extrinsic constant of time. Then, we use it to construct two equivalent fractional embeddings which retains homogeneity. The notion of fractional constant is also discussed through this formalism. Finally, an illustration is given with natural Lagrangian systems, and the case of the harmonic oscillator is entirely treated.
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02.30.Hq Ordinary differential equations
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Energy of hydrodynamic and magnetohydrodynamic waves with point and continuous spectra

M. Hirota and Y. Fukumoto

J. Math. Phys. 49, 083101 (2008); http://dx.doi.org/10.1063/1.2969275 (28 pages) | Cited 8 times

Online Publication Date: 20 August 2008

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Energy of waves (or eigenmodes) in an ideal fluid and plasma is formulated in the noncanonical Hamiltonian context. By imposing the kinematical constraint on perturbations, the linearized Hamiltonian equation provides a formal definition of wave energy not only for eigenmodes corresponding to point spectra but also for singular ones corresponding to a continuous spectrum. The latter becomes dominant when mean fields have inhomogeneity originating from shear or gradient of the fields. The energy of each wave is represented by the eigenfrequency multiplied by the wave action, which is nothing but the action variable and, moreover, is associated with a derivative of a suitably defined dispersion relation. The sign of the action variable is crucial to the occurrence of Hopf bifurcation in Hamiltonian systems of finite degrees of freedom [ M. G. Krein, Dokl. Akad. Nauk SSSR, Ser. A 73, 445 (1950) ]. Krein’s idea is extended to the case of coalescence between point and continuous spectra.
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47.35.Tv Magnetohydrodynamic waves
52.35.Bj Magnetohydrodynamic waves (e.g., Alfven waves)
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Analysis of periodic Schrödinger operators: Regularity and approximation of eigenfunctions

Eugenie Hunsicker, Victor Nistor, and Jorge O. Sofo

J. Math. Phys. 49, 083501 (2008); http://dx.doi.org/10.1063/1.2957940 (21 pages) | Cited 2 times

Online Publication Date: 4 August 2008

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Let V be a real valued potential that is smooth everywhere on math3, except at a periodic, discrete set S of points, where it has singularities of the Coulomb-type Z/r. We assume that the potential V is periodic with period lattice L. We study the spectrum of the Schrödinger operator H = −Δ+V acting on the space of Bloch waves with arbitrary, but fixed, wavevector k. Let math ≔ math3/L. Let u be an eigenfunction of H with eigenvalue λ and let ϵ>0 be arbitrarily small. We show that the classical regularity of the eigenfunction u is u ∊ H5/2−ϵ(math) in the usual Sobolev spaces, and u ∊ K3/2−ϵm(math\S) in the weighted Sobolev spaces. The regularity index m can be as large as desired, which is crucial for numerical methods. For any choice of the Bloch wavevector k, we also show that H has compact resolvent and hence a complete eigenfunction expansion. The case of the hydrogen atom suggests that our regularity results are optimal. We present two applications to the numerical approximation of eigenvalues: using wave functions and using piecewise polynomials.
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03.65.Ge Solutions of wave equations: bound states
02.10.Ud Linear algebra
03.65.Db Functional analytical methods
03.65.Fd Algebraic methods
02.60.-x Numerical approximation and analysis
02.10.De Algebraic structures and number theory

An extended procedure for finding exact solutions of partial differential equations arising from potential symmetries. Applications to gas dynamics

Alexei F. Cheviakov

J. Math. Phys. 49, 083502 (2008); http://dx.doi.org/10.1063/1.2956502 (18 pages) | Cited 3 times

Online Publication Date: 7 August 2008

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Lie point symmetries and nonlocal symmetries of partial differential equation (PDE) systems are widely used for construction of exact invariant solutions. In this paper we describe an extended algorithmic procedure that, for a given nonlocal (potential) symmetry, can yield additional exact solutions, which cannot be found using the usual algorithm. In particular, such additional solutions are exact solutions of the given PDE system, but are not invariant solutions of the corresponding potential system. As an example, we consider a tree of nonlocally related PDE systems for Lagrange planar gas dynamics equations and classify its nonlocal symmetries for an ideal polytropic gas. For two different nonlocal symmetries of the Lagrange system, we demonstrate that the extended method yields wider classes of exact solutions than the usual method.
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05.60.-k Transport processes
02.20.Qs General properties, structure, and representation of Lie groups
02.30.Jr Partial differential equations
)(ν/π)nvol(mathn/Γ). Furthermore, we show that the eigenspace EΓ,χν(0) associated with the lowest Landau level of mathν is isomorphic to the space, OΓ,χν(mathn), of holomorphic functions on mathn satisfying g(z+γ) = χ(γ)eν/2|γ|2+ν〈z,γ〉g(z),(∗) that we can realize also as the null space of the differential operator, ∑j = 1n(−∂2/∂zj∂mathj+νmathj∂/∂mathj) acting on C∞ functions on mathn satisfying (∗).

Landau automorphic functions on Cn of magnitude ν

A. Ghanmi and A. Intissar

J. Math. Phys. 49, 083503 (2008); http://dx.doi.org/10.1063/1.2958090 (20 pages)

Online Publication Date: 8 August 2008

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We investigate the spectral theory of the invariant Landau Hamiltonian, mathν = −1/2{4∑j = 1n∂2/∂zj∂mathj+2ν∑j = 1n(zj∂/∂zj−mathj∂/∂mathj)−ν2|z|2}, acting on the space FΓ,χν of (Γ,χ)-automorphic functions on mathn, constituted of C∞ functions satisfying the functional equation f(z+γ) = χ(γ)mathf(z);  z ∊ mathn,γ ∊ Γ, for given real number ν>0, lattice Γ of mathn and a map χ:Γ→U(1) such that the triplet (ν,Γ,χ) satisfies a Riemann–Dirac quantization-type condition. More precisely, we show that the eigenspace EΓ,χν(λ) = {f ∊ FΓ,χν; mathνf = ν(2λ+n)f}; λ ∊ math, is nontrivial if and only if λ = l = 0,1,2,…,. In such case, EΓ,χν(l) is a finite dimensional vector space whose the dimension is given explicitly by dim EΓ,χν(l) = (
n+l−1
l
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03.65.Db Functional analytical methods
03.65.Fd Algebraic methods
02.10.Ud Linear algebra
02.30.Sa Functional analysis
02.30.Tb Operator theory

Asymptotic theory of the linear transport equation in anisotropic media

Richard Sanchez, Jean Ragusa, and Emiliano Masiello

J. Math. Phys. 49, 083504 (2008); http://dx.doi.org/10.1063/1.2966094 (18 pages)

Online Publication Date: 13 August 2008

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We consider linear transport in an anisotropic medium with velocity dependent cross sections σ(r,v,t) and scattering kernel P(r,v′→v,t). We introduce a scaling in terms of a small parameter ϵ, where the leading-order term describes an equilibrium in velocity space between collisions with a cross section that is an even function of v and scattering modes even-even and odd-odd in v and v′. We show that the asymptotic solution of the transport equation leads to a diffusion equation with a drift term with an error in ϵ2 and derive consistent initial and boundary conditions from the analysis of the initial and boundary layers. The analysis of the drift terms shows that they result from anisotropic interactions with the medium and also from streaming between neighboring but different equilibria. The restriction of our results to isotropic media yields back the Larsen–Keller diffusion equation, while the one-speed form reduces to the result obtained by Pomraning and Prinja [Ann. Nucl. Energy 22, 159 (1995)] for the particular case of isotropic cross sections with an “output” scattering kernel P(r,Ω,t).
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05.60.-k Transport processes

Plane gravitational waves of gauge-invariant generalized field equations with asymmetric fundamental tensor in plane symmetry

S. D. Katore and R. S. Rane

J. Math. Phys. 49, 083505 (2008); http://dx.doi.org/10.1063/1.2966078 (13 pages)

Online Publication Date: 18 August 2008

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Buchdahl [Q. J. Math. 8, 89 (1957) and 9, 257 (1958)] has proposed the study of gauge-invariant generalization of field theories with asymmetric fundamental tensor. In this paper, the solution of plane gravitational waves in these Buchdahl's field theories have been investigated by the authors in plane symmetric space-time. Plane symmetry was studied and defined by Taub [Ann. Math. 53, 472 (1951)] . It has been shown that these solutions become identical with those of strong field equations of Einstein’s nonsymmetric unified field theory in plane symmetry under certain cases.
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04.30.-w Gravitational waves
04.25.D- Numerical relativity

Inverse scattering method and soliton double solution family for the general symplectic gravity model

Ya-Jun Gao

J. Math. Phys. 49, 083506 (2008); http://dx.doi.org/10.1063/1.2957941 (11 pages)

Online Publication Date: 19 August 2008

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A previously established Hauser–Ernst-type extended double-complex linear system is slightly modified and used to develop an inverse scattering method for the stationary axisymmetric general symplectic gravity model. The reduction procedures in this inverse scattering method are found to be fairly simple, which makes the inverse scattering method applied fine and effective. As an application, a concrete family of soliton double solutions for the considered theory is obtained.
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11.10.Lm Nonlinear or nonlocal theories and models
11.25.-w Strings and branes
04.20.Jb Exact solutions
04.65.+e Supergravity

Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives

Thabet Maraaba (Abdeljawad), Dumitru Baleanu, and Fahd Jarad

J. Math. Phys. 49, 083507 (2008); http://dx.doi.org/10.1063/1.2970709 (11 pages) | Cited 4 times

Online Publication Date: 20 August 2008

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The existence and uniqueness theorems for functional right-left delay and left-right advanced fractional functional differential equations with bounded delay and advance, respectively, are proved. The continuity with respect to the initial function for these equations is also proved under some Lipschitz kind conditions. The Q-operator is used to transform the delay-type equation to an advanced one and vice versa. An example is given to clarify the results.
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02.30.Ks Delay and functional equations
02.30.Tb Operator theory
02.30.Hq Ordinary differential equations

Solutions for confluent and double-confluent Heun equations

Léa Jaccoud El-Jaick and Bartolomeu D. B. Figueiredo

J. Math. Phys. 49, 083508 (2008); http://dx.doi.org/10.1063/1.2970150 (28 pages) | Cited 3 times

Online Publication Date: 25 August 2008

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This paper examines some solutions for confluent and double-confluent Heun equations. In the first place, we review two Leaver’s solutions in series of regular and irregular confluent hypergeometric functions for the confluent equation [ Leaver, E. W., J. Math. Phys. 27, 1238 (1986) ] and introduce an additional expansion in series of irregular confluent hypergeometric functions. Then, we find the conditions under which one of these solutions can be written as a linear combination of the others. In the second place, by means of limiting procedures we generate solutions for the double-confluent equation as well as for special limits of both the confluent and double-confluent equations. Finally, we present some problems which are ruled by one or the other of these four equations.
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02.60.Lj Ordinary and partial differential equations; boundary value problems

Symplectic and multisymplectic Lobatto methods for the “good” Boussinesq equation

A. Aydın and B. Karasözen

J. Math. Phys. 49, 083509 (2008); http://dx.doi.org/10.1063/1.2970148 (18 pages) | Cited 4 times

Online Publication Date: 27 August 2008

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In this paper, we construct second order symplectic and multisymplectic integrators for the “good” Boussineq equation using the two-stage Lobatto IIIA-IIIB partitioned Runge–Kutta method, which yield an explicit scheme and is equivalent to the classical central difference approximation to the second order spatial derivative. Numerical dispersion properties and the stability of both integrators are investigated. Numerical results for different solitary wave solutions confirm the excellent long time behavior of symplectic and multisymplectic integrators by preserving local and global energy and momentum.
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05.45.Yv Solitons
02.60.-x Numerical approximation and analysis
02.60.Lj Ordinary and partial differential equations; boundary value problems

Quaternionic and Poisson–Lie structures in three-dimensional gravity: The cosmological constant as deformation parameter

C. Meusburger and B. J. Schroers

J. Math. Phys. 49, 083510 (2008); http://dx.doi.org/10.1063/1.2973040 (27 pages) | Cited 5 times

Online Publication Date: 29 August 2008

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Each of the local isometry groups arising in three-dimensional (3d) gravity can be viewed as a group of unit (split) quaternions over a ring which depends on the cosmological constant. In this paper we explain and prove this statement and use it as a unifying framework for studying Poisson structures associated with the local isometry groups. We show that, in all cases except for the case of Euclidean signature with positive cosmological constant, the local isometry groups are equipped with the Poisson–Lie structure of a classical double. We calculate the dressing action of the factor groups on each other and find, among others, a simple and unified description of the symplectic leaves of SU(2) and SL(2,math). We also compute the Poisson structure on the dual Poisson–Lie groups of the local isometry groups and on their Heisenberg doubles; together, they determine the Poisson structure of the phase space of 3d gravity in the so-called combinatorial description.
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98.80.Cq Particle-theory and field-theory models of the early Universe (including cosmic pancakes, cosmic strings, chaotic phenomena, inflationary universe, etc.)
95.30.Sf Relativity and gravitation
11.30.Ly Other internal and higher symmetries
04.50.-h Higher-dimensional gravity and other theories of gravity
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Publisher's Note: “Purely squeezed states for quantum deformed systems” [ J. Math. Phys. 49, 062104 (2008) ]

A. N. F. Aleixo and A. B. Balantekin

J. Math. Phys. 49, 089901 (2008); http://dx.doi.org/10.1063/1.2977472 (1 page)

Online Publication Date: 20 August 2008

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Abstract Unavailable
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99.10.Fg Publisher's note
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