Showing papers in "Journal of Mathematical Physics in 2006"
TL;DR: In this article, the authors proposed a completely integrable wave equation: mt+mx(u2−ux2)+2m2ux=0, m=u−uxx.
Abstract: In this paper, we propose a new completely integrable wave equation: mt+mx(u2−ux2)+2m2ux=0, m=u−uxx. The equation is derived from the two dimensional Euler equation and is proven to have Lax pair and bi-Hamiltonian structures. This equation possesses new cusp solitons—cuspons, instead of regular peakons ce−∣x−ct∣ with speed c. Through investigating the equation, we develop a new kind of soliton solutions—“W/M”-shape-peaks solitons. There exist no smooth solitons for this integrable water wave equation.
328 citations
TL;DR: In this paper, it was shown that on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states, and the Clifford group is identified as the set of unitary operations which preserve positivity.
Abstract: We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary operations which preserve positivity. The result can be seen as a discrete version of Hudson’s theorem. Hudson established that for continuous variable systems, the Wigner function of a pure state has no negative values if and only if the state is Gaussian. Turning to mixed states, it might be surmised that only convex combinations of stabilizer states give rise to non-negative Wigner distributions. We refute this conjecture by means of a counterexample. Further, we give an axiomatic characterization which completely fixes the definition of the Wigner function and compare two approaches to stabilizer states for Hilbert spaces of prime-power dimensions. In the course of the discussion, we derive explicit formulas for the number of stabilizer codes defined on such systems.
307 citations
TL;DR: In this article, the fractional Schrodinger equation was solved for a free particle and for an infinite square potential well, and the energy levels and the normalized wave functions of a particle in a potential well were obtained.
Abstract: The fractional Schrodinger equation is solved for a free particle and for an infinite square potential well. The fundamental solution of the Cauchy problem for a free particle, the energy levels and the normalized wave functions of a particle in a potential well are obtained. In the barrier penetration problem, the reflection coefficient and transmission coefficient of a particle from a rectangular potential wall is determined. In the quantum scattering problem, according to the fractional Schrodinger equation, the Green’s function of the Lippmann-Schwinger integral equation is given.
231 citations
TL;DR: In this article, a relation between semidirect sums of Lie algebras and integrable couplings of lattice equations is established, and a practicable way to construct integrably couplings is further proposed.
Abstract: A relation between semidirect sums of Lie algebras and integrable couplings of lattice equations is established, and a practicable way to construct integrable couplings is further proposed. An application of the resulting general theory to the generalized Toda spectral problem yields two classes of integrable couplings for the generalized Toda hierarchy of lattice equations. The construction of integrable couplings using semidirect sums of Lie algebras provides a good source of information on complete classification of integrable lattice equations.
196 citations
TL;DR: In this article, the minimum time population transfer problem for the z component of the spin of a (spin 1/2) particle, driven by a magnetic field, that is constant along the z axis and controlled along the x axis, with bounded amplitude, is considered.
Abstract: In this paper we consider the minimum time population transfer problem for the z component of the spin of a (spin 1/2) particle, driven by a magnetic field, that is constant along the z axis and controlled along the x axis, with bounded amplitude. On the Bloch sphere (i.e., after a suitable Hopf projection), this problem can be attacked with techniques of optimal syntheses on two-dimensional (2-D) manifolds. Let (−E,E) be the two energy levels, and ∣Ω(t)∣≤M the bound on the field amplitude. For each couple of values E and M, we determine the time optimal synthesis starting from the level −E, and we provide the explicit expression of the time optimal trajectories, steering the state one to the state two, in terms of a parameter that can be computed solving numerically a suitable equation. For M∕E≪1, every time optimal trajectory is bang-bang and, in particular, the corresponding control is periodic with frequency of the order of the resonance frequency ωR=2E. On the other side, for M∕E>1, the time optimal ...
193 citations
TL;DR: In this paper, the authors propose an outgrowth of the expansion method introduced by de Azcarraga et al. The basic idea consists in considering the direct product between an Abelian semigroup S and a Lie algebra g. General conditions under which relevant subalgebras can systematically be extracted from S×g are given.
Abstract: We propose an outgrowth of the expansion method introduced by de Azcarraga et al. [Nucl. Phys. B 662, 185 (2003)]. The basic idea consists in considering the direct product between an Abelian semigroup S and a Lie algebra g. General conditions under which relevant subalgebras can systematically be extracted from S×g are given. We show how, for a particular choice of semigroup S, the known cases of expanded algebras can be reobtained, while new ones arise from different choices. Concrete examples, including the M algebra and a D’Auria-Fre-like superalgebra, are considered. Finally, we find explicit, nontrace invariant tensors for these S-expanded algebras, which are essential ingredients in, e.g., the formulation of supergravity theories in arbitrary space-time dimensions.
188 citations
TL;DR: This paper studies the further relations among Tsallis type entropies which are typical nonadditive Entropies, and shows parametrically extended results based on information theory.
Abstract: A chain rule and a subadditivity for the entropy of type β, which is one of the nonadditive entropies, were derived by Daroczy. In this paper, we study the further relations among Tsallis type entropies which are typical nonadditive entropies. The chain rule is generalized by showing it for Tsallis relative entropy and the nonadditive entropy. We show some inequalities related to Tsallis entropies, especially the strong subadditivity for Tsallis type entropies and the subadditivity for the nonadditive entropies. The subadditivity and the strong subadditivity naturally lead to define Tsallis mutual entropy and Tsallis conditional mutual entropy, respectively, and then we show again chain rules for Tsallis mutual entropies. We give properties of entropic distances in terms of Tsallis entropies. Finally we show parametrically extended results based on information theory.
167 citations
TL;DR: In this article, the authors provide a complete picture of contractivity of trace preserving positive maps with respect to p-norms and show that for p>1 contractivity holds in general if and only if the map is unital.
Abstract: We provide a complete picture of contractivity of trace preserving positive maps with respect to p-norms. We show that for p>1 contractivity holds in general if and only if the map is unital. When the domain is restricted to the traceless subspace of Hermitian matrices, then contractivity is shown to hold in the case of qubits for arbitrary p⩾1 and in the case of qutrits if and only if p=1, ∞. In all noncontractive cases best possible bounds on the p-norms are derived.
162 citations
TL;DR: In this paper, the inverse scattering transform for the vector defocusing nonlinear Schrodinger (NLS) equation with nonvanishing boundary values at infinity is constructed, and the discrete spectrum, bound states and symmetries of the direct problem are discussed.
Abstract: The inverse scattering transform for the vector defocusing nonlinear Schrodinger (NLS) equation with nonvanishing boundary values at infinity is constructed. The direct scattering problem is formulated on a two-sheeted covering of the complex plane. Two out of the six Jost eigenfunctions, however, do not admit an analytic extension on either sheet of the Riemann surface. Therefore, a suitable modification of both the direct and the inverse problem formulations is necessary. On the direct side, this is accomplished by constructing two additional analytic eigenfunctions which are expressed in terms of the adjoint eigenfunctions. The discrete spectrum, bound states and symmetries of the direct problem are then discussed. In the most general situation, a discrete eigenvalue corresponds to a quartet of zeros (poles) of certain scattering data. The inverse scattering problem is formulated in terms of a generalized Riemann-Hilbert (RH) problem in the upper/lower half planes of a suitable uniformization variable. Special soliton solutions are constructed from the poles in the RH problem, and include dark-dark soliton solutions, which have dark solitonic behavior in both components, as well as dark-bright soliton solutions, which have one dark and one bright component. The linear limit is obtained from the RH problem and is shown to correspond to the Fourier transform solution obtained from the linearized vector NLS system.
151 citations
TL;DR: In this paper, a discussion of character formulas for positive energy, unitary irreducible representations of the conformal group is given, employing Verma modules and Weyl group reflections.
Abstract: A discussion of character formulas for positive energy, unitary irreducible representations of the conformal group is given, employing Verma modules and Weyl group reflections. Product formulas for various conformal group representations are found. These include generalizations of those found by Flato and Fronsdal for SO(3,2). In even dimensions the products for free representations split into two types depending on whether the dimension is divisible by four or not.
135 citations
TL;DR: In this article, a general class of line-soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation was studied by investigating the Wronskian form of its tau-function.
Abstract: We study a general class of line-soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation by investigating the Wronskian form of its tau-function. We show that, in addition to the previously known line soliton solutions of KPII, this class also contains a large variety of multisoliton solutions, many of which exhibit nontrivial spatial interaction patterns. We also show that, in general, such solutions consist of unequal numbers of incoming and outgoing line solitons. From the asymptotic analysis of the tau function, we explicitly characterize the incoming and outgoing line solitons of this class of solutions. We illustrate these results by discussing several examples.
TL;DR: In this paper, a simple exact analytical solution of the relativistic Duffin-Kemmer-Petiau equation within the framework of the asymptotic iteration method is presented.
Abstract: A simple exact analytical solution of the relativistic Duffin-Kemmer-Petiau equation within the framework of the asymptotic iteration method is presented. Exact bound state energy eigenvalues and corresponding eigenfunctions are determined for the relativistic harmonic oscillator as well as the Coulomb potentials. As a nontrivial example, the anharmonic oscillator is solved and the energy eigenvalues are obtained within the perturbation theory using the asymptotic iteration method.
TL;DR: A connection between the Yang-Baxter relation for maps and the multidimensional consistency property of integrable equations on quad-graphs is investigated in this article, based on the symmetry analysis of the corresponding equations.
Abstract: A connection between the Yang-Baxter relation for maps and the multidimensional consistency property of integrable equations on quad-graphs is investigated. The approach is based on the symmetry analysis of the corresponding equations. It is shown that the Yang-Baxter variables can be chosen as invariants of the multiparameter symmetry groups of the equations. We use the classification results by Adler, Bobenko, and Suris to demonstrate this method. Some new examples of Yang-Baxter maps are derived in this way from multifield integrable equations.
TL;DR: In this paper, a structure and classification theory of second-order superintegrable systems, both classical and quantum, in conformally flat spaces is presented, where the classical algebra always closes at order 6 and each classical system has a unique quantum extension.
Abstract: This paper is the conclusion of a series that lays the groundwork for a structure and classification theory of second-order superintegrable systems, both classical and quantum, in conformally flat spaces. For two-dimensional and for conformally flat three-dimensional spaces with nondegenerate potentials we have worked out the structure of the classical systems and shown that the quadratic algebra always closes at order 6. Here we describe the quantum analogs of these results. We show that, for nondegenerate potentials, each classical system has a unique quantum extension. We also correct an error in an earlier paper in the series (that does not alter the structure results) and we elucidate the distinction between superintegrable systems with bases of functionally linearly independent and functionally linearly dependent symmetries.
TL;DR: In this article, the Stackel transform is used as an invertible mapping between classical superintegrable systems on different three-dimensional spaces, and it is shown that all systems with nondegenerate potentials are multiseparable.
Abstract: This article is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems, both classical and quantum, in conformally flat spaces. In the first part of the article we study the Stackel transform (or coupling constant metamorphosis) as an invertible mapping between classical superintegrable systems on different three-dimensional spaces. We show first that all superintegrable systems with nondegenerate potentials are multiseparable and then that each such system on any conformally flat space is Stackel equivalent to a system on a constant curvature space. In the second part of the article we classify all the superintegrable systems that admit separation in generic coordinates. We find that there are eight families of these systems.
TL;DR: In this paper, the authors studied the global spectrum fluctuations for β-Hermite and β-Laguerre ensembles via tridiagonal matrix models and proved that the fluctuations describe a Gaussian process on polynomials.
Abstract: We study the global spectrum fluctuations for β-Hermite and β-Laguerre ensembles via the tridiagonal matrix models introduced previously by the present authors [J. Math. Phys. 43, 5830 (2002)], and prove that the fluctuations describe a Gaussian process on polynomials. We extend our results to slightly larger classes of random matrices.
TL;DR: In this article, a Lagrangian field theory of elastic bodies with evolving reference configurations is developed, where the reference configuration and the ambient space are modeled as Riemannian manifolds with their own metrics.
Abstract: This paper presents some developments related to the idea of covariance in elasticity. The geometric point of view in continuum mechanics is briefly reviewed. Building on this, regarding the reference configuration and the ambient space as Riemannian manifolds with their own metrics, a Lagrangian field theory of elastic bodies with evolving reference configurations is developed. It is shown that even in this general setting, the Euler-Lagrange equations resulting from horizontal referential variations are equivalent to those resulting from vertical spatial variations. The classical Green-Naghdi-Rivilin theorem is revisited and a material version of it is discussed. It is shown that energy balance, in general, cannot be invariant under isometries of the reference configuration, which in this case is identified with a subset of R 3 . Transformation properties of balance of energy under rigid translations and rotations of the reference configuration is obtained. The spatial covariant theory of elasticity is also revisited. The transformation of balance of energy under an arbitrary diffeomorphism of the reference configuration is obtained and it is shown that some nonstandard terms appear in the transformed balance of energy. Then conditions under which energy balance is materially covariant are obtained. It is seen that material covariance of energy balance is equivalent to conservation of mass, isotropy, material Doyle-Ericksen formula and an extra condition that we call configurational inviscidity. In the last part of the paper, the connection between Noether’s theorem and covariance is investigated. It is shown that the DoyleEricksen formula can be obtained as a consequence of spatial covariance of Lagrangian density. Similarly, it is shown that the material Doyle-Ericksen formula can be obtained from material covariance of Lagrangian density. © 2006 American Institute of Physics. DOI: 10.1063/1.2190827
TL;DR: In this article, the exact solution of the Klein-Gordon equation in the presence of noncentral equal scalar and vector potentials was obtained by using Nikiforov-Uvarov method.
Abstract: We present an alternative and simple method for the exact solution of the Klein-Gordon equation in the presence of the noncentral equal scalar and vector potentials by using Nikiforov-Uvarov method. The exact bound state energy eigenvalues and corresponding eigenfunctions are obtained for a particle bound in a potential of V(r,θ)=α∕r+β∕(r2sin2θ)+γcosθ∕(r2sin2θ) type.
TL;DR: In this paper, it was shown that the two-dimensional superintegrable systems with quadratic integrals of motion on a manifold can be classified by using the Poisson algebra of the integral of motion.
Abstract: In this paper we prove that the two-dimensional superintegrable systems with quadratic integrals of motion on a manifold can be classified by using the Poisson algebra of the integrals of motion. There are six general fundamental classes of superintegrable systems. Analytic formulas for the involved integrals are calculated in all the cases. All the known superintegrable systems are classified as special cases of these six general classes.
TL;DR: In this paper, an uncertainty Fisher information relation in quantum mechanics is derived for multidimensional single-particle systems with central potentials, based on the concept of Fisher information in the two complementary position and momentum spaces, which is a gradient functional of the corresponding probability distributions.
Abstract: An uncertainty Fisher information relation in quantum mechanics is derived for multidimensional single-particle systems with central potentials. It is based on the concept of Fisher information in the two complementary position and momentum spaces, which is a gradient functional of the corresponding probability distributions. The lower bound of the product of position and momentum Fisher informations is shown to depend on the orbital and magnetic quantum numbers of the physical state and the space dimensionality. Applications to various elementary systems is discussed.
TL;DR: The annihilation-creation operators a (±) are defined as the positive/negative frequency parts of the exact Heisenberg operator solution for thesinusoidal coordinate as discussed by the authors, and the relative weights of various terms in them are solely determined by the energy spectrum.
Abstract: The annihilation-creation operators a (±) are defined as the positive/negative frequency parts of the exact Heisenberg operator solution for the ‘sinusoidal coordinate’. Thus a (±) are hermitian conjugate to each other and the relative weights of various terms in them are solely determined by the energy spectrum. This unified method applies to most of the solvable quantum mechanics of single degree of freedom including those belonging to the ‘discrete’ quantum mechanics.
TL;DR: All desirable features with correct signs for the relevant terms are obtained uniquely and without any fine tuning in the spectral action of the noncommutative space defined by the standard model.
Abstract: The arbitrary mass scale in the spectral action for the Dirac operator is made dynamical by introducing a dilaton field. We evaluate all the low-energy terms in the spectral action and determine the dilaton couplings. These results are applied to the spectral action of the noncommutative space defined by the standard model. We show that the effective action for all matter couplings is scale invariant, except for the dilaton kinetic term and Einstein-Hilbert term. The resulting action is almost identical to the one proposed for making the standard model scale invariant as well as the model for extended inflation and has the same low-energy limit as the Randall-Sundrum model. Remarkably, all desirable features with correct signs for the relevant terms are obtained uniquely and without any fine tuning.
TL;DR: In this paper, a generalized second-order nonlinear ordinary differential equation (ODE) of the form x+(k1xq+k2)x+k3x2q+1+k4xq +1+λ1x=0, where ki's, i=1,2,3,4, λ1, and q are arbitrary parameters, is considered.
Abstract: In this paper, we consider a generalized second-order nonlinear ordinary differential equation (ODE) of the form x+(k1xq+k2)x+k3x2q+1+k4xq+1+λ1x=0, where ki’s, i=1,2,3,4, λ1, and q are arbitrary parameters, which includes several physically important nonlinear oscillators such as the simple harmonic oscillator, anharmonic oscillator, force-free Helmholtz oscillator, force-free Duffing and Duffing–van der Pol oscillators, modified Emden-type equation and its hierarchy, generalized Duffing–van der Pol oscillator equation hierarchy, and so on, and investigate the integrability properties of this rather general equation. We identify several new integrable cases for arbitrary value of the exponent q,q∊R. The q=1 and q=2 cases are analyzed in detail and the results are generalized to arbitrary q. Our results show that many classical integrable nonlinear oscillators can be derived as subcases of our results and significantly enlarge the list of integrable equations that exists in the contemporary literature. T...
TL;DR: In this article, the variance and Laplace transform of the probability law of linear eigenvalue statistics of unitary invariant matrix models of n×n Hermitian matrices as n→∞ were studied.
Abstract: We study the variance and the Laplace transform of the probability law of linear eigenvalue statistics of unitary invariant Matrix Models of n×n Hermitian matrices as n→∞. Assuming that the test function of statistics is smooth enough and using the asymptotic formulas by Deift et al. [ Commun. Pure Appl. Math.52, 1325–1425 (1999)] for orthogonal polynomials with varying weights, we show first that if the support of the Density of States of the model consists of q≥2 intervals, then in the global regime the variance of statistics is a quasiperiodic function of n as n→∞ generically in the potential, determining the model. We show next that the exponent of the Laplace transform of the probability law is not, in general, 1∕2× variance, as it should be if the Central Limit Theorem would be valid, and we find the asymptotic form of the Laplace transform of the probability law in certain cases.
TL;DR: Theoretical background of continuous contractions of finite-dimensional Lie algebras is rigorously formulated and developed in this article, where necessary criteria of contractions are collected and new criteria are proposed.
Abstract: Theoretical background of continuous contractions of finite-dimensional Lie algebras is rigorously formulated and developed. In particular, known necessary criteria of contractions are collected and new criteria are proposed. A number of requisite invariant and semi-invariant quantities are calculated for wide classes of Lie algebras including all low-dimensional Lie algebras. An algorithm that allows one to handle one-parametric contractions is presented and applied to low-dimensional Lie algebras. As a result, all one-parametric continuous contractions for both the complex and real Lie algebras of dimensions not greater than 4 are constructed with intensive usage of necessary criteria of contractions and with studying correspondence between real and complex cases. Levels and colevels of low-dimensional Lie algebras are discussed in detail. Properties of multiparametric and repeated contractions are also investigated.
TL;DR: The generalized quantum entropies are introduced in this paper and some important properties such as nonnegativity, continuity, and concavity are proved.
Abstract: The generalized quantum entropies are introduced in this paper. Some important properties such as nonnegativity, continuity, and concavity are proved. But different from the Von Neumann entropy, the subadditivity and additivity fail for the quantum unified (r,s)-entropy in general.
TL;DR: In this paper, Bogomolnyi, Prasad, and Sommerfeld (BPS) and non-BPS solutions of the Yang-Mills equations on the noncommutative space Rθ2n×S2 which have manifest spherical symmetry were constructed using SU(2)-equivariant dimensional reduction techniques.
Abstract: We construct explicit Bogomolnyi, Prasad, Sommerfeld (BPS) and non-BPS solutions of the Yang-Mills equations on the noncommutative space Rθ2n×S2 which have manifest spherical symmetry. Using SU(2)-equivariant dimensional reduction techniques, we show that the solutions imply an equivalence between instantons on Rθ2n×S2 and non-Abelian vortices on Rθ2n, which can be interpreted as a blowing-up of a chain of D0-branes on Rθ2n into a chain of spherical D2-branes on Rθ2n×S2. The low-energy dynamics of these configurations is described by a quiver gauge theory which can be formulated in terms of new geometrical objects generalizing superconnections. This formalism enables the explicit assignment of D0-brane charges in equivariant K-theory to the instanton solutions.
TL;DR: In this paper, the fractional Hamiltonian analysis of 1+1 dimensional field theory is investigated and fractional Ostrogradski's formulation is obtained, where the classical results are obtained when fractional derivatives are replaced with the integer order derivatives.
Abstract: The fractional Hamiltonian analysis of 1+1 dimensional field theory is investigated and the fractional Ostrogradski’s formulation is obtained. The fractional path integral of both simple harmonic oscillator with an acceleration-squares part and a damped oscillator are analyzed. The classical results are obtained when fractional derivatives are replaced with the integer order derivatives.
TL;DR: In this article, the Riesz fractional derivative of the continuous limit of discrete systems with long-range interactions is defined and the map of discrete models into continuous medium models is defined.
Abstract: Continuous limits of discrete systems with long-range interactions are considered. The map of discrete models into continuous medium models is defined. A wide class of long-range interactions that give the fractional equations in the continuous limit is discussed. The one-dimensional systems of coupled oscillators for this type of long-range interactions are considered. The discrete equations of motion are mapped into the continuum equation with the Riesz fractional derivative.
TL;DR: In this paper, it was shown that any chiral set of fermions can be embedded in a larger set of Fermions which is chiral and anomaly-free.
Abstract: We present new techniques for finding anomaly-free sets of fermions. Although the anomaly cancellation conditions typically include cubic equations with integer variables that cannot be solved in general, we prove by construction that any chiral set of fermions can be embedded in a larger set of fermions which is chiral and anomaly-free. Applying these techniques to extensions of the standard model, we find anomaly-free models that have arbitrary quark and lepton charges under an additional U(1) gauge group.