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Mar 2011

Volume 52, Issue 3, Articles (03xxxx)

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back to top Quantum Mechanics (General and Nonrelativistic)

Intrinsic structure of state space of a quantum system

Zhi-Hao Ma and Sen Zhu

J. Math. Phys. 52, 032101 (2011); http://dx.doi.org/10.1063/1.3559133 (11 pages)

Online Publication Date: 2 March 2011

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Quantum states play a fundamental role in quantum physics; so it is necessary to study intrinsic structure of quantum states. In this paper, we study topological structure and measurable structure of state space of a quantum system, and find that almost all physical important properties on quantum states coincide.
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03.65.-w Quantum mechanics
02.40.Pc General topology

Noncommutative oscillators from a Hopf algebra twist deformation. A first principles derivation

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

J. Math. Phys. 52, 032102 (2011); http://dx.doi.org/10.1063/1.3562510 (16 pages) | Cited 3 times

Online Publication Date: 9 March 2011

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Noncommutative oscillators are first-quantized through an abelian Drinfel'd twist deformation of a Hopf algebra and its action on a module. Several important and subtle issues making the quantization possible are solved. The spectrum of the single-particle Hamiltonians is computed. The multiparticle Hamiltonians are fixed, unambiguously, by the Hopf algebra coproduct. The symmetry under particle exchange is guaranteed. In d = 2 dimensions the rotational invariance is preserved, while in d = 3 the so(3) rotational invariance is broken down to an so(2) invariance.
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03.65.Ge Solutions of wave equations: bound states
02.10.Ud Linear algebra
03.65.-w Quantum mechanics

On the Cauchy problem for Gross–Pitaevskii hierarchies

Zeqian Chen and Chuangye Liu

J. Math. Phys. 52, 032103 (2011); http://dx.doi.org/10.1063/1.3567168 (13 pages)

Online Publication Date: 14 March 2011

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The purpose of this paper is to investigate the Cauchy problem for the Gross–Pitaevskii infinite linear hierarchy of equations on mathn, n ⩾ 1. We prove local existence and uniqueness of solutions in certain Sobolev-type spaces Hξα of sequences of marginal density operators with α > n/2. In particular, we give a clear discussion of all cases α > n/2, which covers the local well-posedness problem for the Gross–Pitaevskii hierarchy in this situation.
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03.75.-b Matter waves
02.30.-f Function theory, analysis

Information entropy of conditionally exactly solvable potentials

D. Dutta and P. Roy

J. Math. Phys. 52, 032104 (2011); http://dx.doi.org/10.1063/1.3566977 (7 pages)

Online Publication Date: 16 March 2011

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We evaluate Shannon entropy for the position and momentum eigenstates of some conditionally exactly solvable potentials which are isospectral to harmonic oscillator and whose solutions are given in terms of exceptional orthogonal polynomials. The Bialynicki–Birula–Mycielski inequality has also been tested for a number of states.
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05.70.Ce Thermodynamic functions and equations of state
03.65.Ge Solutions of wave equations: bound states
02.10.De Algebraic structures and number theory

A non-Abelian SO(8) monopole as generalization of Dirac-Yang monopoles for a 9-dimensional space

Van-Hoang Le and Thanh-Son Nguyen

J. Math. Phys. 52, 032105 (2011); http://dx.doi.org/10.1063/1.3567422 (11 pages) | Cited 1 time

Online Publication Date: 18 March 2011

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We establish an explicit form of a non-Abelian SO(8) monopole in a 9-dimensional space and show that it is indeed a direct generalization of Dirac and Yang monopoles. Using the generalized Hurwitz transformation, we have found a connection between a 16-dimensional harmonic oscillator and a 9-dimensional hydrogenlike atom in the field of the SO(8) monopole (MICZ-Kepler problem). Using the built connection the group of dynamical symmetry of the 9-dimensional MICZ-Kepler problem is found as SO(10, 2).
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11.30.Ly Other internal and higher symmetries
02.20.Qs General properties, structure, and representation of Lie groups
11.15.-q Gauge field theories
03.65.Ge Solutions of wave equations: bound states

Transition representations of quantum evolution with application to scattering resonances

Y. Strauss

J. Math. Phys. 52, 032106 (2011); http://dx.doi.org/10.1063/1.3559003 (28 pages) | Cited 1 time

Online Publication Date: 24 March 2011

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A Lyapunov operator is a self-adjoint quantum observable whose expectation value varies monotonically as time increases and may serve as a marker for the flow of time in a quantum system. In this paper it is shown that the existence of a certain type of Lyapunov operator leads to representations of the quantum dynamics, termed transition representations, in which an evolving quantum state ψ(t) is decomposed into a sum ψ(t) = ψb(t) + ψf(t) of a backward asymptotic component and a forward asymptotic component such that the evolution process is represented as a transition from ψb(t) to ψf(t). When applied to the evolution of scattering resonances, such transition representations separate the process of decay of a scattering resonance from the evolution of outgoing waves corresponding to the probability “released” by the resonance and carried away to spatial infinity. This separation property clearly exhibits the spatial probability distribution profile of a resonance. Moreover, it leads to the definition of exact resonance states as elements of the physical Hilbert space corresponding to the scattering problem. These resonance states evolve naturally according to a semigroup law of evolution.
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03.65.-w Quantum mechanics
03.65.Ge Solutions of wave equations: bound states
02.50.Cw Probability theory

Exact diagonalization of 1D interacting spinless Fermions

Heiner Kohler

J. Math. Phys. 52, 032107 (2011); http://dx.doi.org/10.1063/1.3563580 (24 pages) | Cited 1 time

Online Publication Date: 31 March 2011

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We acquire a method of constructing an infinite set of exact eigenfunctions of 1D interacting spinless Fermionic systems. Creation and annihilation operators for the interacting system are found and thereby the many-body Hamiltonian is diagonalized. The formalism is applied to several examples. One example is the theory of Jack polynomials. For the Calogero–Moser–Sutherland Hamiltonian a direct proof is given that the asymptotic Bethe ansatz is correct.
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05.30.Fk Fermion systems and electron gas
02.10.Ud Linear algebra
back to top Quantum Information and Computation

On quantum network coding

Avinash Jain, Massimo Franceschetti, and David A. Meyer

J. Math. Phys. 52, 032201 (2011); http://dx.doi.org/10.1063/1.3555801 (18 pages)

Online Publication Date: 4 March 2011

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We study the problem of error-free multiple unicast over directed acyclic networks in a quantum setting. We provide a new information-theoretic proof of the known result that network coding does not achieve a larger quantum information flow than what can be achieved by routing for two-pair communication on the butterfly network. We then consider a k-pair multiple unicast problem and for all k ⩾ 2 we show that there exists a family of networks where quantum network coding achieves k-times larger quantum information flow than what can be achieved by routing. Finally, we specify a graph-theoretic sufficient condition for the quantum information flow of any multiple unicast problem to be bounded by the capacity of any sparsest multicut of the network.
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03.67.Dd Quantum cryptography and communication security
03.67.Hk Quantum communication

Decoherence for positive semigroups on M2(math)

Raffaella Carbone, Emanuela Sasso, and Veronica Umanità

J. Math. Phys. 52, 032202 (2011); http://dx.doi.org/10.1063/1.3560478 (17 pages)

Online Publication Date: 21 March 2011

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We extend Blanchard and Olkiewicz's definition of decoherence to open quantum systems whose dynamics are described by semigroups of positive (and not necessarily completely positive) operators on B(h). In particular, in the case h = math2, we completely characterize the decomposition B(h) = M1⊕M2 of B(h) in the sum of a decoherence-free part M1 and of a space M2 on which the semigroup vanishes with time.
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03.65.Db Functional analytical methods
03.65.Fd Algebraic methods
03.65.Aa Quantum systems with finite Hilbert space
02.20.Uw Quantum groups
02.10.-v Logic, set theory, and algebra
back to top Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

Canonical quantization of lattice Higgs–Maxwell–Chern–Simons fields: Osterwalder–Schrader positivity

Daniel A. Bowman and John L. Challifour

J. Math. Phys. 52, 032301 (2011); http://dx.doi.org/10.1063/1.3559122 (20 pages)

Online Publication Date: 1 March 2011

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A Euclidean representation is given for a canonically quantized relativistic Maxwell–Chern–Simons field on a lattice, which approximates a complex measure on a space of distributions. Using a path-space formula for the nonself-adjoint Hamiltonian, the relation between Euclidean Osterwalder–Schrader positivity, the Krein metric, and Gauss’ law is examined.
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11.15.Yc Chern-Simons gauge theory
11.10.Cd Axiomatic approach
11.15.Ha Lattice gauge theory

Laplacians in polar matrix coordinates and radial fermionization in higher dimensions

Mthokozisi Masuku and João P. Rodrigues

J. Math. Phys. 52, 032302 (2011); http://dx.doi.org/10.1063/1.3553456 (10 pages) | Cited 1 time

Online Publication Date: 4 March 2011

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We consider the quantum mechanical Hamiltonian of two, space indexed, Hermitian matrices. By introducing matrix valued polar coordinates, we obtain the form of the Laplacian acting on invariant states. For potentials depending only on the eigenvalues of the radial matrix, we establish that the radially invariant sector is equivalent to a system of noninteracting 2 + 1 dimensional fermions and obtain its density description. For a larger number of matrices, the presence of a repulsive radial intereigenvalue potential is identified.
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03.65.Fd Algebraic methods
02.10.Ud Linear algebra

Toward an axiomatic formulation of noncommutative quantum field theory

M. Chaichian, M. N. Mnatsakanova, K. Nishijima, A. Tureanu, and Yu. S. Vernov 

J. Math. Phys. 52, 032303 (2011); http://dx.doi.org/10.1063/1.3567411 (13 pages) | Cited 1 time

Online Publication Date: 31 March 2011

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We propose new Wightman functions as vacuum expectation values of products of field operators in the noncommutative space–time. These Wightman functions involve the ⋆-product among the fields, compatible with the twisted Poincaré symmetry of the noncommutative quantum field theory (NC QFT). In the case of only space–space noncommutativity (θ0i = 0), we prove the CPT theorem using the noncommutative form of the Wightman functions. We also show that the spin-statistics theorem, demonstrated for the simplest case of a scalar field, holds in NC QFT within this formalism.
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11.10.Nx Noncommutative field theory
11.30.Er Charge conjugation, parity, time reversal, and other discrete symmetries
11.30.Cp Lorentz and Poincaré invariance
11.10.Cd Axiomatic approach
11.30.Ly Other internal and higher symmetries
back to top General Relativity and Gravitation

Angular momentum at null infinity in five dimensions

Kentaro Tanabe, Norihiro Tanahashi, and Tetsuya Shiromizu

J. Math. Phys. 52, 032501 (2011); http://dx.doi.org/10.1063/1.3559917 (17 pages)

Online Publication Date: 3 March 2011

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In this paper, using the Bondi coordinates, we discuss the angular momentum at null infinity in five dimensions and address the Poincare covariance of the Bondi mass and angular momentum. We also show the angular momentum loss/gain law due to gravitational waves. In four dimensions, the angular momentum at null infinity has the supertranslational ambiguity and then it is known that we cannot construct well-defined angular momentum there. On the other hand, we would stress that we can define angular momentum at null infinity without any ambiguity in higher dimensions. This is because of the nonexistence of supertranslations in higher dimensions.
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04.30.-w Gravitational waves
04.50.-h Higher-dimensional gravity and other theories of gravity

The motion of a body in Newtonian theories

James Owen Weatherall

J. Math. Phys. 52, 032502 (2011); http://dx.doi.org/10.1063/1.3556608 (16 pages) | Cited 1 time

Online Publication Date: 10 March 2011

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A theorem due to Bob Geroch and Pong Soo Jang [J. Math. Phys. 16(1), 65 (1975)] provides the sense in which the geodesic principle has the status of a theorem in general relativity (GR). Here we show that a similar theorem holds in the context of geometrized Newtonian gravitation (often called Newton–Cartan theory). It follows that in Newtonian gravitation, as in GR, inertial motion can be derived from other central principles of the theory.
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04.20.-q Classical general relativity
02.40.Hw Classical differential geometry
95.30.Sf Relativity and gravitation
back to top Dynamical Systems

Sufficient conditions for a nondegenerate Hopf bifurcation in a generalized Lagrange–Poisson problem

Juan L. G. Guirao and Juan A. Vera

J. Math. Phys. 52, 032701 (2011); http://dx.doi.org/10.1063/1.3559064 (11 pages)

Online Publication Date: 1 March 2011

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In this paper we provide sufficient conditions for the existence of a nondegenerate Hamiltonian Hopf bifurcation at the equilibria corresponding to the rotation around the vertical axis of a symmetric gyrostat with a fixed point under the effect of an axially symmetric potential.
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02.30.Oz Bifurcation theory
05.45.-a Nonlinear dynamics and chaos

Existence of dark solitons in a class of stationary nonlinear Schrödinger equations with periodically modulated nonlinearity and periodic asymptotics

J. Belmonte-Beitia and J. Cuevas

J. Math. Phys. 52, 032702 (2011); http://dx.doi.org/10.1063/1.3559118 (9 pages)

Online Publication Date: 1 March 2011

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In this paper, we give a proof of the existence of stationary dark soliton solutions or heteroclinic orbits of nonlinear equations of Schrödinger type with periodic inhomogeneous nonlinearity. The result is illustrated with examples of dark solitons for cubic and photorefractive nonlinearities.
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05.45.Yv Solitons
03.65.Ge Solutions of wave equations: bound states

Darbouxian integrals for generalized Raychaudhuri equations

Claudia Valls

J. Math. Phys. 52, 032703 (2011); http://dx.doi.org/10.1063/1.3559065 (13 pages) | Cited 1 time

Online Publication Date: 2 March 2011

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We give a complete characterization of the Darbouxian first integrals of a generalized Raychaudhuri equation which appears in modern string cosmology and which has the form math = −½x2−αx−2(y2+z2−w2)−2β,math = −(α+x)y−γ,math = −(α+x)z−δ,math = −(α+x)w, where α, β, γ, δ are real parameters. Our approach uses the Darboux theory of integrability.
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02.30.Rz Integral equations

On the constrained B-type Kadomtsev–Petviashvili hierarchy: Hirota bilinear equations and Virasoro symmetry

Hsin-Fu Shen and Ming-Hsien Tu

J. Math. Phys. 52, 032704 (2011); http://dx.doi.org/10.1063/1.3559081 (21 pages) | Cited 1 time

Online Publication Date: 4 March 2011

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We derive the bilinear equations of the constrained BKP hierarchy from the calculus of pseudodifferential operators. The full hierarchy equations can be expressed in Hirota's bilinear form characterized by the functions ρ, σ, and τ. Besides, we also give a modification of the original Orlov–Schulman additional symmetry to preserve the constrained form of the Lax operator for this hierarchy. The vector fields associated with the modified additional symmetry turn out to satisfy a truncated centerless Virasoro algebra.
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03.65.Pm Relativistic wave equations
02.30.Hq Ordinary differential equations

d’Alembert–Lagrange analytical dynamics for nonholonomic systems

M. R. Flannery

J. Math. Phys. 52, 032705 (2011); http://dx.doi.org/10.1063/1.3559128 (29 pages) | Cited 1 time

Online Publication Date: 18 March 2011

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The d’Alembert–Lagrange principle (DLP) is designed primarily for dynamical systems under ideal geometric constraints. Although it can also cover linear-velocity constraints, its application to nonlinear kinematic constraints has so far remained elusive, mainly because there is no clear method whereby the set of linear conditions that restrict the virtual displacements can be easily extracted from the equations of constraint. On recognition that the commutation rule traditionally accepted for velocity displacements in Lagrangian dynamics implies displaced states that do not satisfy the kinematic constraints, we show how the property of possible displaced states can be utilized ab initio so as to provide an appropriate set of linear auxiliary conditions on the displacements, which can be adjoined via Lagrange's multipliers to the d’Alembert–Lagrange equation to yield the equations of state, and also new transpositional relations for nonholonomic systems. The equations of state so obtained for systems under general nonlinear velocity and acceleration constraints are shown to be identical with those derived (in Appendix A) from the quite different Gauss principle. The present advance therefore solves a long outstanding problem on the application of DLP to ideal nonholonomic systems and, as an aside, provides validity to axioms as the Chetaev rule, previously left theoretically unjustified. A more general transpositional form of the Boltzmann–Hamel equation is also obtained.
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45.05.+x General theory of classical mechanics of discrete systems
05.45.-a Nonlinear dynamics and chaos
back to top Classical Mechanics and Classical Fields

Periodic orbits and nonintegrability of generalized classical Yang–Mills Hamiltonian systems

Lidia Jiménez–Lara and Jaume Llibre

J. Math. Phys. 52, 032901 (2011); http://dx.doi.org/10.1063/1.3559145 (9 pages) | Cited 3 times

Online Publication Date: 2 March 2011

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The averaging theory of first order is applied to study a generalized Yang–Mills system with two parameters. Two main results are proved. First, we provide sufficient conditions on the two parameters of the generalized system to guarantee the existence of continuous families of isolated periodic orbits parameterized by the energy, and these families are given up to first order in a small parameter. Second, we prove that for the nonintegrable classical Yang–Mills Hamiltonian systems, in the sense of Liouville–Arnold, which have the isolated periodic orbits found with averaging theory, cannot exist in any second first integral of class C1. This is important because most of the results about integrability deals with analytic or meromorphic integrals of motion.
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03.65.Ge Solutions of wave equations: bound states
11.15.-q Gauge field theories
02.30.Jr Partial differential equations

Wake potentials and impedances of charged beams in gradually tapering structures

D. A. Burton, D. C. Christie, J. D. A. Smith, and R. W. Tucker

J. Math. Phys. 52, 032902 (2011); http://dx.doi.org/10.1063/1.3559157 (30 pages)

Online Publication Date: 4 March 2011

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We develop an analytical method for calculating the geometric wakefield and impedances of an ultrarelativistic beam propagating on- and off-axis through an axially symmetric geometry with slowly varying circular cross-section, such as a transition. Unlike previous analytical methods, our approach permits detailed perturbative investigation of geometric wakefields, and detailed perturbative investigation of impedance as a function of frequency. We compare the accuracy of the results of our approach with numerical simulations performed using the code ECHO and determine parameters in which there is good agreement with our asymptotic analysis.
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29.27.Eg Beam handling; beam transport
29.27.Fh Beam characteristics

Polynomial constants of motion for Calogero-type systems in three dimensions

Claudia Chanu, Luca Degiovanni, and Giovanni Rastelli

J. Math. Phys. 52, 032903 (2011); http://dx.doi.org/10.1063/1.3559132 (7 pages) | Cited 3 times

Online Publication Date: 8 March 2011

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We give an explicit and concise formula for higher degree polynomial first integrals of a family of Calogero-type Hamiltonian systems in dimension three. These first integrals, together with the already known ones, prove the maximal superintegrability of the systems.
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02.30.Rz Integral equations
back to top Fluids

Nonuniform dependence for the Cauchy problem of the general b-equation

Yan Li

J. Math. Phys. 52, 033101 (2011); http://dx.doi.org/10.1063/1.3553184 (14 pages)

Online Publication Date: 11 March 2011

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This paper is concerned with the nonuniform dependence on initial data for the general b-equation. We prove that the solution map of the Cauchy problem of the b-equation is not uniformly continuous in Hs(math), s > 3/2.
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02.30.-f Function theory, analysis

A regularity criterion for the three-dimensional nematic liquid crystal flow in terms of one directional derivative of the velocity

Qiao Liu, Jihong Zhao, and Shangbin Cui

J. Math. Phys. 52, 033102 (2011); http://dx.doi.org/10.1063/1.3567170 (8 pages)

Online Publication Date: 17 March 2011

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In this paper we provide a sufficient condition for regularity of solutions to the 3D nematic liquid crystal flow in the entire space in terms of one directional derivative of the velocity field. More precisely, we prove that if ∂3u belongs to Lβ(0,T;Lα(math3)) with math+math ≤ 1 and α > 3, then the solution (u, d) is regular.
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47.57.Lj Flows of liquid crystals
02.30.-f Function theory, analysis
42.70.Df Liquid crystals

Lie symmetry analysis and exact solutions of the quasigeostrophic two-layer problem

Alexander Bihlo and Roman O. Popovych

J. Math. Phys. 52, 033103 (2011); http://dx.doi.org/10.1063/1.3567175 (24 pages)

Online Publication Date: 29 March 2011

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The quasigeostrophic two-layer model is of superior interest in dynamic meteorology since it is one of the easiest ways to study baroclinic processes in geophysical fluid dynamics. The complete set of point symmetries of the two-layer equations is determined. An optimal set of one- and two-dimensional inequivalent subalgebras of the maximal Lie invariance algebra is constructed. On the basis of these subalgebras, we exhaustively carry out group-invariant reduction and compute various classes of exact solutions. Wherever possible, reference to the physical meaning of the exact solutions is given. In particular, the well-known baroclinic Rossby wave solutions in the two-layer model are rediscovered.
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92.60.Bh General circulation
02.20.Sv Lie algebras of Lie groups
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