Volume 49, Issue 8, August 2008
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Minimal length uncertainty relations and new shape invariant models
View Description Hide DescriptionThis paper identifies a new class of shape invariant models. These models are based on extensions of conventional quantum mechanics that satisfy a stringmotivated 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.

Extending the class of solvable potentials. I. The infinite potential well with a sinusoidal bottom
View Description Hide DescriptionThis 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 threeterm 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 wellknown energy spectra and the corresponding wave functions. Moreover, some of the new solvable problems show evidence of a Klauderlike phenomenon. In the present work, we give an exact solution for the infinite potential well with a bottom that has a sinusoidal shape.

On the epistemic view of quantum states
View Description Hide DescriptionWe 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.

Fisher information of special functions and secondorder differential equations
View Description Hide DescriptionWe investigate a basic question of analytic information theory, namely, the evaluation of the Fisher information and the relative Fisher information with respect to a nonnegative function, for the probability distributions obtained by squaring the special functions of mathematical physics which are solutions of secondorder differential equations. We obtain explicit expressions for these informationtheoretic properties via the expectation values of the coefficients of the differential equation. We illustrate our approach for various nonrelativistic dimensional wavefunctions and some special functions of physicomathematical interest. Emphasis is made in the Nikiforov–Uvarov hypergeometrictype functions, which include and generalize the Hermite functions and the Gauss and Kummer hypergeometric functions, among others.

Geometric phase for nonHermitian Hamiltonians and its holonomy interpretation
View Description Hide DescriptionFor an arbitrary possibly nonHermitian matrix Hamiltonian that might involve exceptional points, we construct an appropriate parameter space and line bundle over such that the adiabatic geometric phases associated with the eigenstates of the initial Hamiltonian coincide with the holonomies of . We examine the case of 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.

Wigner oscillators, twisted Hopf algebras, and second quantization
View Description Hide DescriptionBy correctly identifying the role of the central extension in the centrally extended Heisenberg algebra, we show that it is indeed possible to construct a Hopf algebraic structure on the corresponding enveloping algebra and eventually deform it through the Drinfeld twist. This Hopf algebraic structure and its deformed version 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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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Asymptotic counting of BPS operators in superconformal field theories
View Description Hide DescriptionWe 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 partition functions. Finally, we discuss the concept of a limit curve arising from the large partition functions, which is related to the notion of a “typical state,” and discuss some implications for the holographic duals.
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 GENERAL RELATIVITY AND GRAVITATION


Rotating black branes in Brans–Dicke–Born–Infeld theory
View Description Hide DescriptionIn this paper, we present a new class of charged rotating black branesolutions 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 branesolutions 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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 DYNAMICAL SYSTEMS


Dispersionful analog of the Whitham hierarchy
View Description Hide DescriptionThe transition from the zerogenus 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 KadomtsevPetviashvili (KP) hierarchy is the Whitham hierarchy. The advantage of the presented approach is the construction of finitefield reductions, which are the main concern in this paper. The theory is illustrated by several significant examples.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Homogeneous fractional embeddings
View Description Hide DescriptionFractional 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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 FLUIDS


Energy of hydrodynamic and magnetohydrodynamic waves with point and continuous spectra
View Description Hide DescriptionEnergy 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. A73, 445 (1950)]. Krein’s idea is extended to the case of coalescence between point and continuous spectra.
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 METHODS OF MATHEMATICAL PHYSICS


Analysis of periodic Schrödinger operators: Regularity and approximation of eigenfunctions
View Description Hide DescriptionLet be a real valued potential that is smooth everywhere on , except at a periodic, discrete set of points, where it has singularities of the Coulombtype . We assume that the potential is periodic with period lattice . We study the spectrum of the Schrödinger operator acting on the space of Bloch waves with arbitrary, but fixed, wavevector . Let . Let be an eigenfunction of with eigenvalue and let be arbitrarily small. We show that the classical regularity of the eigenfunction is in the usual Sobolev spaces, and in the weighted Sobolev spaces. The regularity index can be as large as desired, which is crucial for numerical methods. For any choice of the Bloch wavevector , we also show that 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.

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

Landau automorphic functions on of magnitude
View Description Hide DescriptionWe investigate the spectral theory of the invariant Landau Hamiltonian, , acting on the space of automorphic functions on , constituted of functions satisfying the functional equation, for given real number , lattice of and a map such that the triplet satisfies a Riemann–Dirac quantizationtype condition. More precisely, we show that the eigenspace; , is nontrivial if and only if . In such case, is a finite dimensional vector space whose the dimension is given explicitly by . Furthermore, we show that the eigenspace associated with the lowest Landau level of is isomorphic to the space, , of holomorphic functions on satisfying that we can realize also as the null space of the differential operator, acting on functions on satisfying .

Asymptotic theory of the linear transport equation in anisotropic media
View Description Hide DescriptionWe consider linear transport in an anisotropic medium with velocity dependent cross sections and scattering kernel . We introduce a scaling in terms of a small parameter , where the leadingorder term describes an equilibrium in velocity space between collisions with a cross section that is an even function of and scattering modes eveneven and oddodd in and . We show that the asymptotic solution of the transportequation leads to a diffusionequation with a drift term with an error in 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 diffusionequation, while the onespeed form reduces to the result obtained by Pomraning and Prinja [Ann. Nucl. Energy22, 159 (1995)] for the particular case of isotropic cross sections with an “output” scattering kernel .

Plane gravitational waves of gaugeinvariant generalized field equations with asymmetric fundamental tensor in plane symmetry
View Description Hide DescriptionBuchdahl [Q. J. Math.8, 89 (1957) and 9, 257 (1958)] has proposed the study of gaugeinvariant 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 spacetime. 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.

Inverse scattering method and soliton double solution family for the general symplectic gravity model
View Description Hide DescriptionA previously established Hauser–Ernsttype extended doublecomplex 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.

Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives
View Description Hide DescriptionThe existence and uniqueness theorems for functional rightleft delay and leftright advanced fractional functionaldifferential 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 operator is used to transform the delaytype equation to an advanced one and vice versa. An example is given to clarify the results.

Solutions for confluent and doubleconfluent Heun equations
View Description Hide DescriptionThis paper examines some solutions for confluent and doubleconfluent 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 doubleconfluent equation as well as for special limits of both the confluent and doubleconfluent equations. Finally, we present some problems which are ruled by one or the other of these four equations.

Symplectic and multisymplectic Lobatto methods for the “good” Boussinesq equation
View Description Hide DescriptionIn this paper, we construct second order symplectic and multisymplectic integrators for the “good” Boussineq equation using the twostage Lobatto IIIAIIIB 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.
