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Oct 2008

Volume 49, Issue 10, Articles (10xxxx)

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Announcement: Special issue on “Integrable Quantum Systems and Solvable Statistical Mechanics Models”

Bruno L. Z. Nachtergaele

J. Math. Phys. 49, 100201 (2008); http://dx.doi.org/10.1063/1.3019306 (1 page)

Online Publication Date: 22 October 2008

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Abstract Unavailable
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01.60.+q Biographies, tributes, personal notes, and obituaries
01.10.Cr Announcements, news, and awards
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Asymptotic performance of optimal state estimation in qubit system

Masahito Hayashi and Keiji Matsumoto

J. Math. Phys. 49, 102101 (2008); http://dx.doi.org/10.1063/1.2988130 (33 pages) | Cited 7 times

Online Publication Date: 2 October 2008

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We derive an asymptotic bound for the error of state estimation when we are allowed to use the quantum correlation in the measuring apparatus. It is also proven that this bound can be achieved in any statistical model in the qubit system. Moreover, we show that this bound cannot be attained by any quantum measurement with no quantum correlation in the measuring apparatus except for several specific statistical models. That is, in such a statistical model, the quantum correlation can improve the accuracy of the estimation in an asymptotic setting.
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03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
03.65.Ta Foundations of quantum mechanics; measurement theory
02.50.Cw Probability theory
03.65.Fd Algebraic methods
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The effect of classical noise on a quantum two-level system

Jean-Philippe Aguilar and Nils Berglund

J. Math. Phys. 49, 102102 (2008); http://dx.doi.org/10.1063/1.2988180 (23 pages) | Cited 2 times

Online Publication Date: 3 October 2008

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We consider a quantum two-level system perturbed by classical noise. The noise is implemented as a stationary diffusion process in the off-diagonal matrix elements of the Hamiltonian, representing a transverse magnetic field. We determine the invariant measure of the system and prove its uniqueness. In the case of Ornstein–Uhlenbeck noise, we determine the speed of convergence to the invariant measure. Finally, we determine an approximate one-dimensional diffusion equation for the transition probabilities. The proofs use both spectral-theoretic and probabilistic methods.
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03.65.Ge Solutions of wave equations: bound states
05.60.-k Transport processes
05.40.Ca Noise
02.50.Cw Probability theory
05.40.Jc Brownian motion
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On the divine clockwork: The spectral gap for the correspondence limit of the Nelson diffusion generator for the atomic elliptic state

Richard Durran, Andrew Neate, Aubrey Truman, and Feng-Yu Wang

J. Math. Phys. 49, 102103 (2008); http://dx.doi.org/10.1063/1.2988715 (22 pages) | Cited 2 times

Online Publication Date: 8 October 2008

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The correspondence limit of the atomic elliptic state in three dimensions is discussed in terms of Nelson’s stochastic mechanics. In previous work we have shown that this approach leads to a limiting Nelson diffusion, and here we discuss in detail the invariant measure for this process and show that it is concentrated on the Kepler ellipse in the plane z = 0. We then show that the limiting Nelson diffusion generator has a spectral gap; thereby proving that in the infinite time limit the density for the limiting Nelson diffusion will converge to its invariant measure. We also include a summary of the Cheeger and Poincaré inequalities, both of which are used in our proof of the existence of the spectral gap.
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05.40.Jc Brownian motion
03.65.Ta Foundations of quantum mechanics; measurement theory
02.50.Ey Stochastic processes
05.60.Gg Quantum transport
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The structure of degradable quantum channels

Toby S. Cubitt, Mary Beth Ruskai, and Graeme Smith

J. Math. Phys. 49, 102104 (2008); http://dx.doi.org/10.1063/1.2953685 (27 pages) | Cited 7 times

Online Publication Date: 8 October 2008

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Degradable quantum channels are among the only channels whose quantum and private classical capacities are known. As such, determining the structure of these channels is a pressing open question in quantum information theory. We give a comprehensive review of what is currently known about the structure of degradable quantum channels, including a number of new results as well as alternate proofs of some known results. In the case of qubits, we provide a complete characterization of all degradable channels with two dimensional output, give a new proof that a qubit channel with two Kraus operators is either degradable or anti-degradable, and present a complete description of anti-degradable unital qubit channels with a new proof. For higher output dimensions we explore the relationship between the output and environment dimensions (dB and dE, respectively) of degradable channels. For several broad classes of channels we show that they can be modeled with an environment that is “small” in the sense of ΦC. Such channels include all those with qubit or qutrit output, those that map some pure state to an output with full rank, and all those which can be represented using simultaneously diagonal Kraus operators, even in a non-orthogonal basis. Perhaps surprisingly, we also present examples of degradable channels with “large” environments, in the sense that the minimal dimension dE>dB. Indeed, one can have dE>¼dB2. These examples can also be used to give a negative answer to the question of whether additivity of the coherent information is helpful for establishing additivity for the Holevo capacity of a pair of channels. In the case of channels with diagonal Kraus operators, we describe the subclasses that are complements of entanglement breaking channels. We also obtain a number of results for channels in the convex hull of conjugations with generalized Pauli matrices. However, a number of open questions remain about these channels and the more general case of random unitary channels.
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03.67.Mn Entanglement measures, witnesses, and other characterizations
03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
03.67.Lx Quantum computation architectures and implementations
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Lower spectral branches of a spin-boson model

Nicolae Angelescu, Robert A. Minlos, Jean Ruiz, and Valentin A. Zagrebnov

J. Math. Phys. 49, 102105 (2008); http://dx.doi.org/10.1063/1.2987721 (29 pages) | Cited 2 times

Online Publication Date: 10 October 2008

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We study the structure of the spectrum of a two-level quantum system weakly coupled to a boson field (spin-boson model). Our analysis allows to avoid the cutoff in the number of bosons, if their spectrum is bounded below by a positive constant. We show that, for small coupling constant, the lower part of the spectrum of the spin-boson Hamiltonian contains (one or two) isolated eigenvalues and (one or two) manifolds of atom +1-boson states indexed by the boson momentum q. The dispersion laws and generalized eigenfunctions of the latter are calculated.
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05.30.Jp Boson systems
02.10.Ud Linear algebra
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Adiabatically switched-on electrical bias and the Landauer–Büttiker formula

H. D. Cornean, P. Duclos, G. Nenciu, and R. Purice

J. Math. Phys. 49, 102106 (2008); http://dx.doi.org/10.1063/1.2992839 (20 pages) | Cited 4 times

Online Publication Date: 10 October 2008

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Consider a three dimensional system which looks like a cross connected pipe system, i.e., a small sample coupled to a finite number of leads. We investigate the current running through this system, in the linear response regime, when we adiabatically turn on an electrical bias between leads. The main technical tool is the use of a finite volume regularization, which allows us to define the current coming out of a lead as the time derivative of its charge. We finally prove that in virtually all physically interesting situations, the conductivity tensor is given by a Landauer–Büttiker type formula.
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05.30.-d Quantum statistical mechanics
02.70.Rr General statistical methods
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Additivity and distinguishability of random unitary channels

Bill Rosgen

J. Math. Phys. 49, 102107 (2008); http://dx.doi.org/10.1063/1.2992977 (16 pages) | Cited 3 times

Online Publication Date: 13 October 2008

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A random unitary channel is one that is given by a convex combination of unitary channels. It is shown that the conjectures on the additivity of the minimum output entropy and the multiplicativity of the maximum output p-norm can be equivalently restated in terms of random unitary channels. This is done by constructing a random unitary approximation to a general quantum channel. This approximation can be constructed efficiently, and so it is also applied to the computational problem of distinguishing quantum circuits. It is shown that the problem of distinguishing random unitary circuits is as hard as the problem of distinguishing general mixed-state circuits, which is complete for the class of problems having quantum interactive proof systems.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.50.-r Probability theory, stochastic processes, and statistics
03.65.-w Quantum mechanics
05.70.Ce Thermodynamic functions and equations of state
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Nine theorems on the unification of quantum mechanics and relativity

A. Kryukov

J. Math. Phys. 49, 102108 (2008); http://dx.doi.org/10.1063/1.2996282 (13 pages) | Cited 3 times

Online Publication Date: 16 October 2008

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A mathematical framework that unifies the standard formalisms of special relativity and quantum mechanics is proposed. For this a Hilbert space H of functions of four variables x,t furnished with an additional indefinite inner product invariant under Poincaré transformations is introduced. For a class of functions in H that are well localized in the time variable, the usual formalism of nonrelativistic quantum mechanics is derived. In particular, the interference in time for these functions is suppressed; a motion in H becomes the usual Schrödinger evolution with t as a parameter. The relativistic invariance of the construction is proved. The usual theory of relativity on Minkowski space-time is shown to be “isometrically and equivariantly embedded” into H. That is, classical space-time is isometrically embedded into H, Poincaré transformations have unique extensions to isomorphisms of H, and the embedding commutes with Poincaré transformations.
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03.65.Ge Solutions of wave equations: bound states
03.30.+p Special relativity
02.30.Uu Integral transforms
03.65.Fd Algebraic methods
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Discrete approximation of quantum stochastic models

Luc Bouten and Ramon Van Handel

J. Math. Phys. 49, 102109 (2008); http://dx.doi.org/10.1063/1.3001109 (19 pages)

Online Publication Date: 20 October 2008

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We develop a general technique for proving convergence of repeated quantum interactions to the solution of a quantum stochastic differential equation. The wide applicability of the method is illustrated in a variety of examples. Our main theorem, which is based on the Trotter–Kato theorem, is not restricted to a specific noise model and does not require boundedness of the limit coefficients.
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03.65.Ta Foundations of quantum mechanics; measurement theory
02.50.Ey Stochastic processes
02.30.-f Function theory, analysis
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Classical limit for semirelativistic Hartree systems

Gonca L. Aki, Peter A. Markowich, and Christof Sparber

J. Math. Phys. 49, 102110 (2008); http://dx.doi.org/10.1063/1.3000059 (10 pages) | Cited 1 time

Online Publication Date: 21 October 2008

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We consider the three-dimensional semirelativistic Hartree model for fast quantum mechanical particles moving in a self-consistent field. Under appropriate assumptions on the initial density matrix as a (fully) mixed quantum state we prove by using Wigner transformation techniques that its classical limit yields the well known relativistic Vlasov–Poisson system. The result holds for the case of attractive and repulsive mean-field interactions, with an additional size constraint in the attractive case.
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03.65.Sq Semiclassical theories and applications
02.10.Yn Matrix theory
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The O(1)-Kepler problems

Guowu Meng

J. Math. Phys. 49, 102111 (2008); http://dx.doi.org/10.1063/1.3000062 (8 pages) | Cited 3 times

Online Publication Date: 21 October 2008

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Let n ≥ 2 be an integer. To each irreducible representation σ of O(1), an O(1)-Kepler problem in dimension n is constructed and analyzed. This system is super integrable and when n = 2 it is equivalent to a generalized MICZ (McIntosh-Cisneros-Zwanziger)-Kepler problem in dimension 2. The dynamical symmetry group of this system is math(2n,math) with the Hilbert space of bound states H(σ) being the unitary highest weight representation of math(2n,math) with highest weight
math
which occurs at the rightmost nontrivial reduction point in the Enright–Howe–Wallach classification diagram for the unitary highest weight modules. (Here |σ| = 0 or 1 depending on whether σ is trivial or not.) Furthermore, it is shown that the correspondence σ↔H(σ) is the theta correspondence for dual pair (O(1),Sp(2n,math))⊆Sp(2n,math).
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45.50.Pk Celestial mechanics
02.20.-a Group theory
02.30.Ik Integrable systems
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Weyl quantization of fractional derivatives

Vasily E. Tarasov

J. Math. Phys. 49, 102112 (2008); http://dx.doi.org/10.1063/1.3009533 (6 pages) | Cited 2 times

Online Publication Date: 31 October 2008

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The quantum analogs of the derivatives with respect to coordinates qk and momenta pk are commutators with operators Pk and Qk. We consider quantum analogs of fractional Riemann–Liouville and Liouville derivatives. To obtain the quantum analogs of fractional Riemann–Liouville derivatives, which are defined on a finite interval of the real axis, we use a representation of these derivatives for analytic functions. To define a quantum analog of the fractional Liouville derivative, which is defined on the real axis, we can use the representation of the Weyl quantization by the Fourier transformation.
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03.65.Db Functional analytical methods
02.30.Nw Fourier analysis
02.30.Tb Operator theory
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
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Stochastic quantization for complex actions

G. Menezes and N. F. Svaiter

J. Math. Phys. 49, 102301 (2008); http://dx.doi.org/10.1063/1.2996276 (10 pages) | Cited 2 times

Online Publication Date: 9 October 2008

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We use the stochastic quantization method to study systems with complex valued path integral weights. We assume a Langevin equation with a memory kernel and Einstein’s relations with colored noise. The equilibrium solution of this non-Markovian Langevin equation is analyzed. We show that for a large class of elliptic non-Hermitian operators acting on scalar functions on Euclidean space, which define different models in quantum field theory, converge to an equilibrium state in the asymptotic limit of the Markov parameter τ→∞. Moreover, as we expected, we obtain the Schwinger functions of the theory.
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03.65.Ta Foundations of quantum mechanics; measurement theory
11.10.Cd Axiomatic approach
02.50.Ga Markov processes
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Super Schrödinger algebra in AdS/CFT

Makoto Sakaguchi and Kentaroh Yoshida

J. Math. Phys. 49, 102302 (2008); http://dx.doi.org/10.1063/1.2998205 (13 pages) | Cited 14 times

Online Publication Date: 15 October 2008

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We discuss (extended) super-Schrödinger algebras obtained as subalgebras of the superconformal algebra psu(2,2∣4). The Schrödinger algebra with two spatial dimensions can be embedded into so(4,2). In the superconformal case the embedded algebra may be enhanced to the so-called super-Schrödinger algebra. In fact, we find an extended super-Schrödinger subalgebra of psu(2,2∣4). It contains 24 supercharges (i.e., 3/4 of the original supersymmetries) and the generators of so(6), as well as the generators of the original Schrödinger algebra. In particular, the 24 supercharges come from 16 rigid supersymmetries and half of 16 superconformal ones. Moreover, this superalgebra contains a smaller super-Schrödinger subalgebra, which is a supersymmetric extension of the original Schrödinger algebra and so(6) by eight supercharges (half of 16 rigid supersymmetries). It is still a subalgebra even if there are no so(6) generators. We also discuss super-Schrödinger subalgebras of the superconformal algebras, osp(8∣4) and osp(8∗∣4), and find super Schrödinger subalgebras in the same way.
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11.30.Pb Supersymmetry
11.25.Hf Conformal field theory, algebraic structures
03.65.Ge Solutions of wave equations: bound states
02.40.Re Algebraic topology
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On the embedding of space-time in five-dimensional Weyl spaces

F. Dahia, G. A. T. Gomez, and C. Romero

J. Math. Phys. 49, 102501 (2008); http://dx.doi.org/10.1063/1.3000049 (12 pages) | Cited 3 times

Online Publication Date: 14 October 2008

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We revisit Weyl geometry in the context of recent higher-dimensional theories of space-time. After introducing the Weyl theory in a modern geometrical language we present some results that represent extensions of Riemannian theorems. We consider the theory of local embeddings and submanifolds in the context of Weyl geometries and show how a Riemannian space-time may be locally and isometrically embedded in a Weyl bulk. We discuss the problem of classical confinement and the stability of motion of particles and photons in the neighborhood of branes for the case when the Weyl bulk has the geometry of a warped product space. We show how the confinement and stability properties of geodesics near the brane may be affected by the Weyl field. We construct a classical analog of quantum confinement inspired in theoretical-field models by considering a Weyl scalar field which depends only on the extra coordinate.
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98.80.Jk Mathematical and relativistic aspects of cosmology
02.40.Ky Riemannian geometries
02.40.Sf Manifolds and cell complexes
95.30.Sf Relativity and gravitation
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Analytic integrability of a Chua system

Jaume Llibre and Clàudia Valls

J. Math. Phys. 49, 102701 (2008); http://dx.doi.org/10.1063/1.2992481 (9 pages)

Online Publication Date: 8 October 2008

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We consider the system math = a(z−a1x3−a2x2−bx), math = −z, math = −b1x+y+b2z, where a and b are parameters and b1 = 7/10, b2 = 6/25, a1 = 44/3, and a2 = 41/2. We analyze the existence of local and global analytic first integrals.
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02.30.Rz Integral equations
02.30.Hq Ordinary differential equations
02.10.De Algebraic structures and number theory
02.30.Ik Integrable systems
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On the classification of Darboux integrable chains

Ismagil Habibullin, Natalya Zheltukhina, and Aslı Pekcan

J. Math. Phys. 49, 102702 (2008); http://dx.doi.org/10.1063/1.2992950 (39 pages) | Cited 5 times

Online Publication Date: 9 October 2008

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We study a differential-difference equation of the form tx(n+1) = f(t(n),t(n+1),tx(n)) with unknown t = t(n,x) depending on x and n. The equation is called a Darboux integrable if there exist functions F (called an x-integral) and I (called an n-integral), both of a finite number of variables x,t(n),t(n±1),t(n±2),…,tx(n),txx(n),…, such that DxF = 0 and DI = I, where Dx is the operator of total differentiation with respect to x and D is the shift operator: Dp(n) = p(n+1). The Darboux integrability property is reformulated in terms of characteristic Lie algebras that give an effective tool for classification of integrable equations. The complete list of equations of the form above admitting nontrivial x-integrals is given in the case when the function f is of the special form f(x,y,z) = z+d(x,y).
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02.30.Ik Integrable systems
02.20.Sv Lie algebras of Lie groups
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Approximation of center manifolds on the renormalization group method

Hayato Chiba

J. Math. Phys. 49, 102703 (2008); http://dx.doi.org/10.1063/1.2996290 (11 pages) | Cited 5 times

Online Publication Date: 13 October 2008

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The renormalization group (RG) method for differential equations is one of the perturbation methods for obtaining approximate solutions. This article shows that the RG method is effectual for obtaining an approximate center manifold and an approximate flow on it when applied to equations having a center manifold.
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05.45.-a Nonlinear dynamics and chaos
05.10.Cc Renormalization group methods
02.40.-k Geometry, differential geometry, and topology
02.30.Oz Bifurcation theory
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Hydrodynamic type integrable equations on a segment and a half-line

Metin Gürses, Ismagil Habibullin, and Kostyantyn Zheltukhin

J. Math. Phys. 49, 102704 (2008); http://dx.doi.org/10.1063/1.2993008 (15 pages)

Online Publication Date: 20 October 2008

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The concept of integrable boundary conditions is applied to hydrodynamic type systems. Examples of such boundary conditions for dispersionless Toda systems are obtained. The close relation of integrable boundary conditions with integrable reductions in multifield systems is observed. The problem of consistency of boundary conditions with the Hamiltonian formulation is discussed. Examples of Hamiltonian integrable hydrodynamic type systems on a segment and a semiline are presented.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
02.30.Rz Integral equations
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Lp-uniform attractor for nonautonomous reaction-diffusion equations in unbounded domains

Xingjie Yan and Chengkui Zhong

J. Math. Phys. 49, 102705 (2008); http://dx.doi.org/10.1063/1.3000575 (17 pages)

Online Publication Date: 21 October 2008

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The goal of this paper is to consider the asymptotic behavior of solutions of nonautonomous classical reaction-diffusion equations in unbounded domains with nonlinearity having a polynomial growth of arbitrary order. The existence and structure of a uniform attractor are obtained in the spaces L2(mathn) and Lp(mathn), respectively.
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05.60.-k Transport processes
02.10.Ab Logic and set theory
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The classical point electron in Colombeau’s theory of nonlinear generalized functions

Andre Gsponer

J. Math. Phys. 49, 102901 (2008); http://dx.doi.org/10.1063/1.2982236 (22 pages) | Cited 1 time

Online Publication Date: 1 October 2008

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The electric and magnetic fields of a pole-dipole singularity attributed to a point-electron singularity in the Maxwell field are expressed in a Colombeau algebra of generalized functions. This enables one to calculate dynamical quantities quadratic in the fields which are otherwise mathematically ill-defined: the self-energy (i.e., “mass”), the self-angular momentum (i.e., “spin”), the self-momentum (i.e., “hidden momentum”), and the self-force. While the total self-force and self-momentum are zero, therefore ensuring that the electron singularity is stable, the mass and spin are diverging integrals of δ2-functions. Yet, after renormalization according to standard prescriptions, the expressions for mass and spin are consistent with quantum theory, including the requirement of a gyromagnetic ratio greater than 1. The most striking result, however, is that the electric and magnetic fields differ from the classical monopolar and dipolar fields by δ-function terms which are usually considered as insignificant, while in a Colombeau algebra these terms are precisely the sources of the mechanical mass and spin of the electron singularity.
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03.50.De Classical electromagnetism, Maxwell equations
03.65.Fd Algebraic methods
02.30.-f Function theory, analysis
02.10.De Algebraic structures and number theory
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Ice-water and liquid-vapor phase transitions by a Ginzburg–Landau model

Mauro Fabrizio

J. Math. Phys. 49, 102902 (2008); http://dx.doi.org/10.1063/1.2992478 (13 pages) | Cited 6 times

Online Publication Date: 14 October 2008

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A model for the first order phase transitions as ice-water and liquid-vapor is proposed using the Ginzburg–Landau equation for the order parameter φ. In this model the density ρ is composed of two quantities ρ0 and ρ1 such that 1/ρ = 1/ρ0+1/ρ1, where ρ1 is strictly connected to the order parameter φ. By means of this decomposition, we are able to represent the Andrew diagram without the use of the heuristic van der Waals equation.
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64.70.D- Solid-liquid transitions
64.70.F- Liquid-vapor transitions
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The calculation of the bilinear Boltzmann operator for gas mixtures

Ji-jun Ao and Achilaltu Wang

J. Math. Phys. 49, 103101 (2008); http://dx.doi.org/10.1063/1.3000563 (10 pages) | Cited 1 time

Online Publication Date: 24 October 2008

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The formula of the bilinear Boltzmann integral operator of gas mixtures is obtained. With the use of Bobylev’s transform it can be carried out much easier and an extended formula of more general validity is obtained. The formula can give the eigenvalues and the eigenfunctions of the bilinear Boltzmann integral operator of a gas mixture, which is essential to the solution of the Boltzmann equation.
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51.10.+y Kinetic and transport theory of gases
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Superdiffusivity of asymmetric energy model in dimensions 1 and 2

Cédric Bernardin

J. Math. Phys. 49, 103301 (2008); http://dx.doi.org/10.1063/1.3000580 (20 pages) | Cited 1 time

Online Publication Date: 22 October 2008

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We discuss an asymmetric energy model introduced by Giardina et al. [J. Math. Phys. 48, 033301 (2007)] . This model is expected to belong to the Kardar–Parisi–Zhang (KPZ) class. We obtain lower bounds for the diffusion coefficient. In particular, the diffusion coefficient is diverging in dimensions one and two as it is expected in the KPZ picture.
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05.60.-k Transport processes
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