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Jun 2010

Volume 51, Issue 6, Articles (06xxxx)

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back to top Quantum Mechanics (General and Nonrelativistic)

Localized eigenfunctions in Šeba billiards

J. P. Keating, J. Marklof, and B. Winn

J. Math. Phys. 51, 062101 (2010); http://dx.doi.org/10.1063/1.3393884 (19 pages)

Online Publication Date: 1 June 2010

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We describe some new families of quasimodes for the Laplacian perturbed by the addition of a potential formally described by a Dirac delta function. As an application, we find, under some additional hypotheses on the spectrum, subsequences of eigenfunctions of Šeba billiards that localize around a pair of unperturbed eigenfunctions.
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03.65.Fd Algebraic methods
02.30.Sa Functional analysis
02.30.Tb Operator theory

On the nonlocality of the fractional Schrödinger equation

M. Jeng, S.-L.-Y. Xu, E. Hawkins, and J. M. Schwarz

J. Math. Phys. 51, 062102 (2010); http://dx.doi.org/10.1063/1.3430552 (6 pages) | Cited 6 times

Online Publication Date: 2 June 2010

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A number of papers over the past eight years have claimed to solve the fractional Schrödinger equation for systems ranging from the one-dimensional infinite square well to the Coulomb potential to one-dimensional scattering with a rectangular barrier. However, some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal, preventing the fractional Schrödinger equation from being solved in the usual piecewise fashion. We focus on the one-dimensional infinite square well and show that the purported ground state, which is based on a piecewise approach, is definitely not a solution of the fractional Schrödinger equation for the general fractional parameter α. On a more positive note, we present a solution to the fractional Schrödinger equation for the one-dimensional harmonic oscillator with α = 1.
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03.65.Ge Solutions of wave equations: bound states

Riccati equation and the problem of decoherence

Bartłomiej Gardas

J. Math. Phys. 51, 062103 (2010); http://dx.doi.org/10.1063/1.3442364 (7 pages) | Cited 4 times

Online Publication Date: 21 June 2010

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The block operator matrix theory is used to investigate the problem of a single qubit. We establish a connection between the Riccati operator equation and the possibility of obtaining an exact reduced dynamics for the qubit in question. The model of the half spin particle in the rotating magnetic field coupling with the external environment is discussed. We show that the model defined in such a way can be reduced to a time independent problem.
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03.67.Lx Quantum computation architectures and implementations
03.65.Yz Decoherence; open systems; quantum statistical methods

Group-theoretical approach to Bloch electron in magnetic field problem

Marko Ćosić

J. Math. Phys. 51, 062104 (2010); http://dx.doi.org/10.1063/1.3453601 (15 pages)

Online Publication Date: 21 June 2010

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In this paper magnetic-translation group theory is extended to include full rotational symmetry of Hamiltonian. Proper generalization of small representation and star of the representation concepts are derived. Irreducible representations of magnetic-translation group and magnetic-space group are presented. Correct form of symmetrized basis function is derived, reflecting symmetry of the magnetic-point group. From viewpoint of group theory reduction of Hamiltonian symmetry group caused by magnetic field and splitting of energy levels is investigated.
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03.65.Fd Algebraic methods
03.65.Ge Solutions of wave equations: bound states

Cranking approach for a complex quantum system

V. M. Kolomietz and S. V. Radionov

J. Math. Phys. 51, 062105 (2010); http://dx.doi.org/10.1063/1.3436579 (10 pages)

Online Publication Date: 29 June 2010

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Within random matrix model, the response of a complex quantum system math(q[t]) on time variations in an external parameter q. In the limit of weak coupling of the external parameter to the quantum system, the different dynamical regimes of the diffusion in a space of the occupancies of eigenstates of math(q) are considered. The main focus is made on measuring the role of memory effects in the quantum diffusive dynamics. Possible macroscopic manifestations of the quantum mechanical diffusion are discussed in the context of the cranking approach, where the time variations in the classical parameter q provides the constancy of energy of the total system.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.10.Ud Linear algebra
05.30.-d Quantum statistical mechanics
05.60.-k Transport processes
back to top Quantum Information and Computation

Fast universal quantum computation with railroad-switch local Hamiltonians

Daniel Nagaj

J. Math. Phys. 51, 062201 (2010); http://dx.doi.org/10.1063/1.3384661 (21 pages) | Cited 1 time

Online Publication Date: 2 June 2010

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We present two universal models of quantum computation with a time-independent, frustration-free Hamiltonian. The first construction uses 3-local (qubit) projectors and the second one requires only 2-local qubit-qutrit projectors. We build on Feynman’s Hamiltonian computer idea [ R. Feynman, Optics News 11, 11 (1985) ] and use a railroad-switch-type clock register. The resources required to simulate a quantum circuit with L gates in this model are O(L) small-dimensional quantum systems (qubits or qutrits), a time-independent Hamiltonian composed of O(L) local, constant norm, projector terms, the possibility to prepare computational basis product states, a running time O(L log2 L), and the possibility to measure a few qubits in the computational basis. Our models also give a simplified proof of the universality of 3-local adiabatic quantum computation.
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03.67.Lx Quantum computation architectures and implementations

Separability of N-particle fermionic states for arbitrary partitions

Tsubasa Ichikawa, Toshihiko Sasaki, and Izumi Tsutsui

J. Math. Phys. 51, 062202 (2010); http://dx.doi.org/10.1063/1.3399807 (23 pages)

Online Publication Date: 8 June 2010

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We present a criterion of separability for arbitrary s partitions of N-particle fermionic pure states. We show that, despite the superficial nonfactorizability due to the antisymmetry required by the indistinguishability of the particles, the states which meet our criterion have factorizable correlations for a class of observables which are specified consistently with the states. The separable states and the associated class of observables share an orthogonal structure, whose nonuniqueness is found to be intrinsic to the multipartite separability and leads to the nontransitivity in the factorizability, in general. Our result generalizes the previous result obtained by Ghirardi et al. [J. Stat. Phys. 108, 49 (2002)] for the s = 2 and s = N case.
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05.30.Fk Fermion systems and electron gas
03.67.Mn Entanglement measures, witnesses, and other characterizations
back to top Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

Abelian link invariants and homology

Enore Guadagnini and Francesco Mancarella

J. Math. Phys. 51, 062301 (2010); http://dx.doi.org/10.1063/1.3431031 (15 pages)

Online Publication Date: 7 June 2010

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We consider the link invariants defined by the quantum Chern–Simons field theory with compact gauge group U(1) in a closed oriented 3-manifold M. The relation of the Abelian link invariants with the homology group of the complement of the links is discussed. We prove that, when M is a homology sphere or when a link—in a generic manifold M—is homologically trivial, the associated observables coincide with the observables of the sphere S3. Finally, we show that the U(1) Reshetikhin–Turaev surgery invariant of the manifold M is not a function of the homology group only, nor a function of the homotopy type of M alone.
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11.15.Yc Chern-Simons gauge theory

BRST detour quantization: Generating gauge theories from constraints

D. Cherney, E. Latini, and A. Waldron

J. Math. Phys. 51, 062302 (2010); http://dx.doi.org/10.1063/1.3372732 (27 pages) | Cited 3 times

Online Publication Date: 7 June 2010

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We present the Becchi–Rouet–Stora–Tyutin (BRST) cohomologies of a class of constraint (super) Lie algebras as detour complexes. By interpreting the components of detour complexes as gauge invariances, Bianchi identities, and equations of motion, we obtain a large class of new gauge theories. The pivotal new machinery is a treatment of the ghost Hilbert space designed to manifest the detour structure. Along with general results, we give details for three of these theories which correspond to gauge invariant spinning particle models of totally symmetric, antisymmetric, and Kähler antisymmetric forms. In particular, we give details of our recent announcement of a (p,q)-form Kähler electromagnetism. We also discuss how our results generalize to other special geometries.
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03.65.-w Quantum mechanics
11.15.-q Gauge field theories
11.30.Pb Supersymmetry

U(1)-invariant membranes: The geometric formulation, Abel, and pendulum differential equations

A. A. Zheltukhin and M. Trzetrzelewski

J. Math. Phys. 51, 062303 (2010); http://dx.doi.org/10.1063/1.3430566 (12 pages) | Cited 2 times

Online Publication Date: 10 June 2010

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The geometric approach to study the dynamics of U(1)-invariant membranes is developed. The approach reveals an important role of the Abel nonlinear differential equation of the first type with variable coefficients depending on time and one of the membrane extendedness parameters. The general solution of the Abel equation is constructed. Exact solutions of the whole system of membrane equations in the D = 5 Minkowski space-time are found and classified. It is shown that if the radial component of the membrane world vector is only time dependent, then the dynamics is described by the pendulum equation.
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11.27.+d Extended classical solutions; cosmic strings, domain walls, texture
04.20.Gz Spacetime topology, causal structure, spinor structure
02.30.Hq Ordinary differential equations
11.30.-j Symmetry and conservation laws

Drinfel’d superdoubles and Poisson–Lie T-plurality in low dimensions

Ladislav Hlavatý, Vojtěch Štěpán, and Jan Vysoký

J. Math. Phys. 51, 062304 (2010); http://dx.doi.org/10.1063/1.3449327 (19 pages)

Online Publication Date: 21 June 2010

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Defining the real Lie superalgebra as real Z2-graded vector space we classify real Manin supertriples and Drinfel’d superdoubles of superdimensions (2,2), (4,2), and (2,4). The Drinfel’d doubles of the superdimension (2,2) are then used for construction of the simplest σ-models related by Poisson–Lie T-plurality.
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02.20.Sv Lie algebras of Lie groups
02.30.Jr Partial differential equations
back to top General Relativity and Gravitation

A dynamic correspondence between Bose–Einstein condensates and Friedmann–Lemaître–Robertson–Walker and Bianchi I cosmology with a cosmological constant

Jennie D’Ambroise and Floyd L. Williams

J. Math. Phys. 51, 062501 (2010); http://dx.doi.org/10.1063/1.3429611 (11 pages)

Online Publication Date: 2 June 2010

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In some interesting work of James Lidsey, the dynamics of Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology with positive curvature and a perfect fluid matter source is shown to be modeled in terms of a time-dependent, harmonically trapped Bose–Einstein condensate. In the present work, we extend this dynamic correspondence to both FLRW and Bianchi I cosmologies in arbitrary dimension, especially when a cosmological constant is present.
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98.80.Jk Mathematical and relativistic aspects of cosmology
04.20.Jb Exact solutions
03.75.Nt Other Bose-Einstein condensation phenomena

On asymptotic structure at null infinity in five dimensions

Kentaro Tanabe, Norihiro Tanahashi, and Tetsuya Shiromizu

J. Math. Phys. 51, 062502 (2010); http://dx.doi.org/10.1063/1.3429580 (14 pages) | Cited 2 times

Online Publication Date: 3 June 2010

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We discuss the asymptotic structure of null infinity in five dimensional space-times. Since it is known that the conformal infinity is not useful for odd higher dimensions, we shall employ the coordinate based method such as the Bondi coordinate first introduced in four dimensions. Then we will define the null infinity and identify the asymptotic symmetry. We will also derive the Bondi mass expression and show its conservation law.
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04.20.Ha Asymptotic structure
04.30.-w Gravitational waves
04.20.Gz Spacetime topology, causal structure, spinor structure
back to top Classical Mechanics and Classical Fields

Hysteresis and phase transitions for one-dimensional and three-dimensional models in shape memory alloys

V. Berti, M. Fabrizio, and D. Grandi

J. Math. Phys. 51, 062901 (2010); http://dx.doi.org/10.1063/1.3430573 (13 pages) | Cited 1 time

Online Publication Date: 23 June 2010

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By means of the Ginzburg–Landau theory of phase transitions, we study a nonisothermal model to characterize the austenite-martensite transition in shape memory alloys. In the first part of this paper, the one-dimensional model proposed by Berti et al. [“Phase transitions in shape memory alloys: A non-isothermal Ginzburg-Landau model,” Physica D 239, 95 (2010)] is modified by varying the expression of the free energy. In this way, the description of the phenomenon of hysteresis, typical of these materials, is improved and the related stress-strain curves are recovered. Then, a generalization of this model to the three-dimensional case is proposed and its consistency with the principles of thermodynamics is proven. Unlike other three-dimensional models, the transition is characterized by a scalar valued order parameter φ and the Ginzburg–Landau equation, ruling the evolution of φ, allows us to prove a maximum principle, ensuring the boundedness of φ itself.
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62.20.fg Shape-memory effect; yield stress; superelasticity
81.30.Kf Martensitic transformations
back to top Statistical Physics

On the solvability of two dimensional semigroup gauge theories

Péter Varga

J. Math. Phys. 51, 063301 (2010); http://dx.doi.org/10.1063/1.3419770 (10 pages)

Online Publication Date: 2 June 2010

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We study the solvability of two dimensional semigroup gauge theories by Migdal’s link elimination method. We determine certain conditions that ensure that the partition sum corresponding to the join of two plaquettes depends only on the holonomy around the boundary of the joined plaquettes. These conditions are checked for a few types of semigroups: 0-groups, cyclic, inverse symmetric, and Brandt semigroups.
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11.15.-q Gauge field theories
02.20.-a Group theory

Formulas for joint probabilities for the asymmetric simple exclusion process

Craig A. Tracy and Harold Widom

J. Math. Phys. 51, 063302 (2010); http://dx.doi.org/10.1063/1.3431977 (10 pages) | Cited 4 times

Online Publication Date: 14 June 2010

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In earlier work, the authors [ Tracy, C. A. and Widom, H., “Integral formulas for the asymmetric simple exclusion process,” Commun. Math. Phys. 279, 815 (2008) ] obtained integral formulas for probabilities for a single particle in the asymmetric simple exclusion process. Here, formulas are obtained for joint probabilities for several particles. In the case of a single particle, the derivation here is simpler than the one in the earlier work for one of its main results.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.50.Cw Probability theory
02.30.Rz Integral equations

Multiscaling for systems with a broad continuum of characteristic lengths and times: Structural transitions in nanocomposites

S. Pankavich and P. Ortoleva

J. Math. Phys. 51, 063303 (2010); http://dx.doi.org/10.1063/1.3420578 (16 pages) | Cited 2 times

Online Publication Date: 28 June 2010

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The multiscale approach to N-body systems is generalized to address the broad continuum of long time and length scales associated with collective behaviors. A technique is developed based on the concept of an uncountable set of time variables and of order parameters (OPs) specifying major features of the system. We adopt this perspective as a natural extension of the commonly used discrete set of time scales and OPs which is practical when only a few, widely separated scales exist. The existence of a gap in the spectrum of time scales for such a system (under quasiequilibrium conditions) is used to introduce a continuous scaling and perform a multiscale analysis of the Liouville equation. A functional-differential Smoluchowski equation is derived for the stochastic dynamics of the continuum of Fourier component OPs. A continuum of spatially nonlocal Langevin equations for the OPs is also derived. The theory is demonstrated via the analysis of structural transitions in a composite material, as occurs for viral capsids and molecular circuits.
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05.45.Tp Time series analysis
05.40.Jc Brownian motion
02.50.Ey Stochastic processes
02.60.Nm Integral and integrodifferential equations

New representations of π and Dirac delta using the nonextensive-statistical-mechanics q-exponential function

M. Jauregui and C. Tsallis

J. Math. Phys. 51, 063304 (2010); http://dx.doi.org/10.1063/1.3431981 (9 pages) | Cited 7 times

Online Publication Date: 29 June 2010

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We present a generalization of the representation in plane waves of Dirac delta, δ(x) = (1/2π)∫−∞∞e−ikxdk, namely, δ(x) = [(2−q)/2π]∫−∞∞eq−ikxdk, using the non-extensive-statistical-mechanics q-exponential function, eqix ≡ [1+(1−q)ix]1/(1−q) with e1ix ≡ eix, x being any real number, for real values of q within the interval [1,2[. Concomitantly, with the development of these new representations of Dirac delta, we also present two new families of representations of the transcendental number π. Incidentally, we remark that the q-plane wave form which emerges, namely, eqikx, is normalizable for 1<q<3, in contrast to the standard one, eikx, which is not.
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05.20.-y Classical statistical mechanics

Bifurcation of binary systems with the Onsager mobility

Chun-Hsiung Hsia

J. Math. Phys. 51, 063305 (2010); http://dx.doi.org/10.1063/1.3406383 (12 pages) | Cited 2 times

Online Publication Date: 29 June 2010

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The main objective of this article is to study the effect of the (nonlinear) Onsager mobility to the phase separation of the binary system, using rigorous bifurcation analysis. In particular, a nondimensional parameter K, depending on the molar density u0 of the homogeneous state, and the critical temperature is derived; the sign of this parameter dictates the type of transition. Also, the analysis indicates that the type of the transition, the critical temperature Tc, and the strength of the deviation of the transition solutions from the homogeneous state are all independent of the choices of the Onsager mobility.
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05.45.-a Nonlinear dynamics and chaos
05.70.-a Thermodynamics
back to top Methods of Mathematical Physics

Fate of the Julia set of higher dimensional maps in the integrable limit

Satoru Saito and Noriko Saitoh

J. Math. Phys. 51, 063501 (2010); http://dx.doi.org/10.1063/1.3430554 (17 pages)

Online Publication Date: 2 June 2010

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By studying higher dimensional rational maps, we have shown, in our previous papers, that periodic points of integrable maps with sufficient number of invariants form invariant varieties of periodic points (IVPPs) different for each period. In this paper, we study the transition of a nonintegrable map to an integrable one. In particular, we investigate analytically where the Julia set goes and how it disappears when the map becomes integrable. We show that the behavior of the Julia set is different, depending on whether the map has an unstable variety of fixed point (UVFP), which becomes nonfixed in the integrable limit. If the map does not have an UVFP, all periodic points approach IVPP or fixed points of the integrable map. Otherwise, a large part of periodic points of all periods approach the UVFP and the UVFP itself becomes a variety of indeterminate points unless it disappears from the map. Moreover, we show that a map recovered by the singularity confinement generates the sequence of all IVPPs.
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05.45.-a Nonlinear dynamics and chaos
02.30.Ik Integrable systems
02.60.-x Numerical approximation and analysis

Singular perturbations with boundary conditions and the Casimir effect in the half space

S. Albeverio, G. Cognola, M. Spreafico, and S. Zerbini

J. Math. Phys. 51, 063502 (2010); http://dx.doi.org/10.1063/1.3397551 (38 pages)

Online Publication Date: 2 June 2010

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We study the self-adjoint extensions of a class of nonmaximal multiplication operators with boundary conditions. We show that these extensions correspond to singular rank 1 perturbations (in the sense of Albeverio and Kurasov [Singular Perturbations of Differential Operaters (Cambridge University Press, Cambridge, 2000)] ) of the Laplace operator, namely, the formal Laplacian with a singular delta potential, on the half space. This construction is the appropriate setting to describe the Casimir effect related to a massless scalar field in the flat space-time with an infinite conducting plate and in the presence of a pointlike “impurity.” We use the relative zeta determinant (as defined in the works of Müller [“Relative zeta functions, relative determinants and scattering theory,” Commun. Math. Phys. 192, 309 (1998)] and Spreafico and Zerbini [“Finite temperature quantum field theory on noncompact domains and application to delta interactions,” Rep. Math. Phys. 63, 163 (2009)] ) in order to regularize the partition function of this model. We study the analytic extension of the associated relative zeta function, and we present explicit results for the partition function and for the Casimir force.
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12.20.Ds Specific calculations
11.10.-z Field theory
02.30.Tb Operator theory
02.30.Sa Functional analysis

The generalized Kupershmidt deformation for constructing new integrable systems from integrable bi-Hamiltonian systems

Yuqin Yao and Yunbo Zeng

J. Math. Phys. 51, 063503 (2010); http://dx.doi.org/10.1063/1.3431967 (14 pages) | Cited 2 times

Online Publication Date: 2 June 2010

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Based on the Kupershmidt deformation for any integrable bi-Hamiltonian systems presented by Kupershmidt [Phys. Lett. A 372, 2634 (2008)] , we propose the generalized Kupershmidt deformation to construct new systems from integrable bi-Hamiltonian systems, which provides a nonholonomic perturbation of the bi-Hamiltonian systems. The generalized Kupershmidt deformation is conjectured to preserve integrability. The conjecture is verified in a few representative cases: Korteweg–de Vries (KdV) equation, Boussinesq equation, Jaulent–Miodek equation, and Camassa–Holm equation. For these specific cases, we present a general procedure to convert the generalized Kupershmidt deformation into the integrable Rosochatius deformation of soliton equation with self-consistent sources, then to transform it into a t-type bi-Hamiltonian system. By using this generalized Kupershmidt deformation some new integrable systems are derived. In fact, this generalized Kupershmidt deformation also provides a new method to construct the integrable Rosochatius deformation of soliton equation with self-consistent sources.
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05.45.Yv Solitons

Bihamiltonian structure of the two-component Kadomtsev–Petviashvili hierarchy of type B

Chao-Zhong Wu and Dingdian Xu

J. Math. Phys. 51, 063504 (2010); http://dx.doi.org/10.1063/1.3431971 (15 pages) | Cited 1 time

Online Publication Date: 2 June 2010

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We employ a Lax pair representation of the two-component Kadomtsev–Petviashvili hierarchy of type B and construct its bihamiltonian structure with R-matrix techniques.
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03.65.Ge Solutions of wave equations: bound states

Realizations of 3-Lie algebras

Ruipu Bai, Chengming Bai, and Jinxiu Wang

J. Math. Phys. 51, 063505 (2010); http://dx.doi.org/10.1063/1.3436555 (12 pages) | Cited 4 times

Online Publication Date: 2 June 2010

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3-Lie algebras have close relationships with many important fields in mathematics and mathematical physics. In this paper, we provide a construction of 3-Lie algebras in terms of Lie algebras and certain linear functions. Moreover, with the construction from γ-matrices and two-dimensional extensions of metric Lie algebras, all the complex 3-Lie algebras in dimension ≤ 5 are obtained along this approach. As a special case, we study the structure of the 3-Lie algebras constructed from the general linear Lie algebras with trace forms and prove that they are semisimple and local.
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02.10.Ud Linear algebra
02.30.Sa Functional analysis
02.10.Yn Matrix theory

Link invariants for flows in higher dimensions

Hugo García-Compeán and Roberto Santos-Silva

J. Math. Phys. 51, 063506 (2010); http://dx.doi.org/10.1063/1.3427319 (17 pages) | Cited 1 time

Online Publication Date: 2 June 2010

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Linking numbers in higher dimensions and their generalization including gauge fields are studied in the context of BF theories. The linking numbers associated with n-manifolds with smooth flows generated by divergence-free p-vector fields, endowed with an invariant flow measure, are computed in the context of quantum field theory. They constitute invariants of smooth dynamical systems (for nonsingular flows) and generalize previous proposals of invariants. In particular, they generalize Arnold’s asymptotic Hopf invariant from three to higher dimensions. This invariant is generalized by coupling with a non-Abelian gauge flat connection with nontrivial holonomy. The computation of the asymptotic Jones–Witten invariants for flows is naturally extended to dimension n = 2p+1. Finally, we give a possible interpretation and implementation of these issues in the context of 11-dimensional supergravity and string theory.
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04.65.+e Supergravity
11.25.Wx String and brane phenomenology
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