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Dec 2011

Volume 52, Issue 12, Articles (12xxxx)

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back to top Quantum Mechanics (General and Nonrelativistic)

On the supremum and infimum of bounded quantum observables

Long Long and Shifang Zhang

J. Math. Phys. 52, 122101 (2011); http://dx.doi.org/10.1063/1.3671331 (6 pages) | Cited 1 time

Online Publication Date: 20 December 2011

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Let S(H) be the set of all bounded self-adjoint linear operators on a complex Hilbert space H. In 2006, Gudder [Math. Slovaca 56, 573 (2006)] introduced a new order ≼ on S(H). Since then, the existence conditions and representations of the supremum and infimum of two elements in S(H) with respect to the order ≼ have been intensively studied. Specifically, Li and Sun [J. Math. Phys. 50, 122107 (2009)]10.1063/1.3272542 obtained simpler representations of A ∧ P and A ∨ P, where A∈S(H) and P is an orthogonal projection on H. In this note, we present more intuitive and concise results on A ∨ P and extend the results of Li and Sun to more general cases. Moreover, some applications of our results are given to show that our results are easier to deal with.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Aa Quantum systems with finite Hilbert space
02.30.Tb Operator theory
03.65.Fd Algebraic methods

Extreme covariant quantum observables in the case of an Abelian symmetry group and a transitive value space

Erkka Theodor Haapasalo and Juha-Pekka Pellonpää

J. Math. Phys. 52, 122102 (2011); http://dx.doi.org/10.1063/1.3668317 (23 pages)

Online Publication Date: 21 December 2011

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We represent quantum observables as normalized positive operator valued measures and consider convex sets of observables which are covariant with respect to a unitary representation of a locally compact Abelian symmetry group G. The value space of such observables is a transitive G-space. We characterize the extreme points of covariant observables and also determine the covariant extreme points of the larger set of all quantum observables. The results are applied to position, position difference, and time observables.
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03.65.Fd Algebraic methods
03.65.Ta Foundations of quantum mechanics; measurement theory
02.20.Bb General structures of groups

An alternative approach to Schrödinger equations with a spatially varying mass

M. Çapak and B. Gönül

J. Math. Phys. 52, 122103 (2011); http://dx.doi.org/10.1063/1.3672208 (5 pages) | Cited 1 time

Online Publication Date: 23 December 2011

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Extending the point canonical transformation approach in a manner distinct from the previous ones, we propose a unified approach of generating potentials of all classes having non-constant masses.
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03.65.Ge Solutions of wave equations: bound states

Spectral problems for the Weyl-ordered form of operators mathnmathn

M. Gianfreda and G. Landolfi

J. Math. Phys. 52, 122104 (2011); http://dx.doi.org/10.1063/1.3667207 (21 pages)

Online Publication Date: 28 December 2011

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In this paper, we consider quantization of powers of the ratio between the Hamiltonian coordinates for position and momentum in one-dimensional systems. The domain of the operators consists of square integrable functions over a finite real interval to ensure boundedness and self-adjointness. The spectral problems for the operators that result from using Weyl-ordering are discussed by introducing Fredholm integral operator forms in position representation, and the symmetry of the actions of the parity and time reversal operators on the kernels is discussed. Finally, the general structures and properties of the eigenfunctions and eigenvalues are also derived and analyzed.
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03.65.-w Quantum mechanics
02.30.Rz Integral equations
02.10.Ud Linear algebra

Exact results for finite quantum Hall systems of electrons at filling factor one: Disk geometry

Orion Ciftja

J. Math. Phys. 52, 122105 (2011); http://dx.doi.org/10.1063/1.3672196 (14 pages) | Cited 1 time

Online Publication Date: 29 December 2011

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We obtain exact analytical expressions for the total energy per particle and related quantities corresponding to a finite quantum Hall system of electrons in a disk geometry when filling factor of the Landau level is one. Such exact results apply to finite systems of electrons with an arbitrary number of particles. The reported calculations for finite systems of electrons in a disk geometry complement earlier calculations for finite systems of electrons in a simpler spherical geometry. The results we provide can serve as benchmarks to gauge the accuracy of various theoretical approximations and numerical methods used to study the properties of strongly correlated many-body systems.
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73.43.-f Quantum Hall effects
71.70.Di Landau levels

The propagator of the attractive delta-Bose gas in one dimension

Sylvain Prolhac and Herbert Spohn

J. Math. Phys. 52, 122106 (2011); http://dx.doi.org/10.1063/1.3663431 (17 pages)

Online Publication Date: 30 December 2011

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We consider the quantum δ-Bose gas on the infinite line. For repulsive interactions, Tracy and Widom have obtained an exact formula for the quantum propagator. In our contribution we explicitly perform its analytic continuation to attractive interactions. We also study the connection to the expansion of the propagator in terms of the Bethe ansatz eigenfunctions. Thereby we provide an independent proof of their completeness.
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05.30.Jp Boson systems
02.10.Ud Linear algebra

Prepotential approach to solvable rational extensions of Harmonic Oscillator and Morse potentials

C.-L. Ho

J. Math. Phys. 52, 122107 (2011); http://dx.doi.org/10.1063/1.3671966 (8 pages)

Online Publication Date: 30 December 2011

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We show how the recently discovered solvable rational extensions of Harmonic Oscillator and Morse potentials can be constructed in a direct and systematic way, without the need of supersymmetry, shape invariance, Darboux-Crum, and Darboux-Bäcklund transformations.
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03.65.Ge Solutions of wave equations: bound states
FREE

Relativistic and nonrelativistic bound states of the isotonic oscillator by Nikiforov-Uvarov method

Sameer M. Ikhdair and Ramazan Sever

J. Math. Phys. 52, 122108 (2011); http://dx.doi.org/10.1063/1.3671640 (14 pages) | Cited 1 time

Online Publication Date: 30 December 2011

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A nonpolynomial one-dimensional quantum potential in the form of an isotonic oscillator (harmonic oscillator with a centripetal barrier) is studied. We provide the nonrelativistic bound state energy spectrum En and the wave functions ψn(x) in terms of the associated Laguerre polynomials in the framework of the Nikiforov-Uvarov method. Under the spin and pseudospin symmetric limits, the analytic eigenvalues and the corresponding two-component upper- and lower-spinors of the Dirac particle are obtained in closed form.
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03.65.Ge Solutions of wave equations: bound states
03.65.Pm Relativistic wave equations
02.10.De Algebraic structures and number theory
02.10.Ud Linear algebra
03.65.Fd Algebraic methods
back to top Quantum Information and Computation

Existence of product vectors and their partial conjugates in a pair of spaces

Young-Hoon Kiem, Seung-Hyeok Kye, and Jungseob Lee

J. Math. Phys. 52, 122201 (2011); http://dx.doi.org/10.1063/1.3663835 (10 pages) | Cited 2 times

Online Publication Date: 7 December 2011

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Let D and E be subspaces of the tensor product of the m- and n-dimensional complex spaces, with co-dimensions k and ℓ, respectively. In order to give upper bounds for ranks of entangled edge states with positive partial transposes, we show that if k + ℓ < m + n − 2, then there must exist a product vector in D whose partial conjugate lies in E. If k + ℓ = m + n − 2, then such a product vector may or may not exist depending on k and ℓ.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
FREE

Three-by-three bound entanglement with general unextendible product bases

Łukasz Skowronek

J. Math. Phys. 52, 122202 (2011); http://dx.doi.org/10.1063/1.3663836 (32 pages)

Online Publication Date: 7 December 2011

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We discuss the subject of unextendible product bases with the orthogonality condition dropped and we prove that the lowest rank non-separable positive-partial-transpose states, i.e., states of rank 4 in 3 × 3 systems are always locally equivalent to a projection onto the orthogonal complement of a linear subspace spanned by an orthogonal unextendible product basis. The product vectors in the kernels of the states belong to a non-zero measure subset of all general unextendible product bases, nevertheless, they can always be locally transformed to the orthogonal form. This fully confirms the surprising numerical results recently reported by Leinaas et al. Parts of the paper rely heavily on the use of Bezout's theorem from algebraic geometry.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Fd Algebraic methods
02.40.-k Geometry, differential geometry, and topology
02.10.-v Logic, set theory, and algebra
03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
FREE

Description of rank four entangled states of two qutrits having positive partial transpose

Lin Chen and Dragomir Ž Đoković

J. Math. Phys. 52, 122203 (2011); http://dx.doi.org/10.1063/1.3663837 (27 pages)

Online Publication Date: 7 December 2011

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It is known that some two-qutrit entangled states of rank 4 with positive partial transpose can be built from the unextendible product bases (UPB) [C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Phys. Rev. Lett. 82, 5385 (1999)]. We show that this fact is indeed universal, namely, all such states can be constructed from UPB as conjectured recently by Leinaas, Myrheim, and Sollid. We also classify the five-dimensional subspaces of two qutrits which contain only finitely many product states (up to scalar multiple), and in particular those spanned by an UPB.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.67.Lx Quantum computation architectures and implementations
03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)

Classical and nonclassical randomness in quantum measurements

Douglas Farenick, Sarah Plosker, and Jerrod Smith

J. Math. Phys. 52, 122204 (2011); http://dx.doi.org/10.1063/1.3668081 (26 pages) | Cited 2 times

Online Publication Date: 13 December 2011

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The space POVM H(X) of positive operator-valued probability measures on the Borel sets of a compact (or even locally compact) Hausdorff space X with values in B(H), the algebra of linear operators acting on a d-dimensional Hilbert space H, is studied from the perspectives of classical and nonclassical convexity through a transform Γ that associates any positive operator-valued measure ν with a certain completely positive linear map Γ(ν) of the homogeneous C*-algebra C(X)⊗B(H) into B(H). This association is achieved by using an operator-valued integral in which nonclassical random variables (that is, operator-valued functions) are integrated with respect to positive operator-valued measures and which has the feature that the integral of a random quantum effect is itself a quantum effect. A left inverse Ω for Γ yields an integral representation, along the lines of the classical Riesz representation theorem for linear functionals on C(X), of certain (but not all) unital completely positive linear maps φ:C(X)⊗B(H)→B(H). The extremal and C*-extremal points of POVM H(X) are determined.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Aa Quantum systems with finite Hilbert space
02.50.Cw Probability theory
03.65.Db Functional analytical methods
03.65.Fd Algebraic methods
back to top Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

The graviton propagator in de Donder gauge on de Sitter background

S. P. Miao, N. C. Tsamis, and R. P. Woodard

J. Math. Phys. 52, 122301 (2011); http://dx.doi.org/10.1063/1.3664760 (23 pages) | Cited 3 times

Online Publication Date: 12 December 2011

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We construct the graviton propagator on de Sitter background in exact de Donder gauge. We prove that it must break de Sitter invariance, just like the propagator of the massless, minimally coupled scalar. Our explicit solutions for its two scalar structure functions preserve spatial homogeneity and isotropy so that the propagator can be used within the larger context of inflationary cosmology; however, it is simple to alter the residual symmetry. Because our gauge condition is de Sitter invariant (although no solution for the propagator can be) renormalization should be simpler using this propagator than one based on a noncovariant gauge. It remains to be seen how other computational steps compare.
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98.80.Cq Particle-theory and field-theory models of the early Universe (including cosmic pancakes, cosmic strings, chaotic phenomena, inflationary universe, etc.)
98.80.Jk Mathematical and relativistic aspects of cosmology
04.60.-m Quantum gravity
back to top General Relativity and Gravitation

Lorentzian covering space and homotopy classes

Do-Hyung Kim

J. Math. Phys. 52, 122501 (2011); http://dx.doi.org/10.1063/1.3670398 (4 pages)

Online Publication Date: 20 December 2011

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We analyze structures of covering space over a Lorentzian manifold. By use of this we show that, if a Lorentzian manifold is globally hyperbolic then for any two causally related points p and q, the number of homotopy classes of causal curves from p to q is finite and each of its homotopy classes has a causal geodesic from p to q.
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02.40.Ma Global differential geometry
back to top Dynamical Systems

Dynamically defined measures and equilibrium states

Ivan Werner

J. Math. Phys. 52, 122701 (2011); http://dx.doi.org/10.1063/1.3666020 (17 pages) | Cited 1 time

Online Publication Date: 15 December 2011

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A technique of dynamically defined measures is developed and its relation to the theory of equilibrium states is shown. The technique uses Carathéodory's method and the outer measure introduced in a previous work by I. Werner [Math. Proc. Camb. Phil. Soc. 140(2), 333–347 (2006)10.1017/S0305004105009072]. As an application, equilibrium states for contractive Markov systems [I. Werner, J. London Math. Soc. 71(1), 236–258 (2005)10.1112/S0024610704006088] are obtained.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.50.Ga Markov processes

Invariant algebraic surfaces for the reduced three-wave interaction system

Adam Mahdi and Claudia Valls

J. Math. Phys. 52, 122702 (2011); http://dx.doi.org/10.1063/1.3672193 (10 pages)

Online Publication Date: 30 December 2011

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In this paper, we consider the three-wave interaction system math = γx+δy+z−2y2, math = γy−δx+2xy, math = −2z−2zx, where γ, δ are real parameters. Our main results are the complete characterization of all values of the parameters γ, δ for which the three-wave interaction system admits either invariant algebraic surfaces or algebraic (polynomial or rational) first integrals.
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02.30.Rz Integral equations
02.10.-v Logic, set theory, and algebra
back to top Classical Mechanics and Classical Fields

Goryachev-Chaplygin, Kovalevskaya, and Brdička-Eardley-Nappi-Witten pp-waves spacetimes with higher rank Stäckel-Killing tensors

G. W. Gibbons and C. Rugina

J. Math. Phys. 52, 122901 (2011); http://dx.doi.org/10.1063/1.3664754 (8 pages)

Online Publication Date: 23 December 2011

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Hidden symmetries of the Goryachev-Chaplygin and Kovalevskaya gyrostats spacetimes, as well as the Brdička-Eardley-Nappi-Witten pp-waves are studied. We find out that these spacetimes possess higher rank Stäckel-Killing tensors and that in the case of the pp-wave spacetimes, the symmetry group of the Stäckel-Killing tensors is the well-known Newton-Hooke group.
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04.20.Gz Spacetime topology, causal structure, spinor structure
02.10.Ud Linear algebra
back to top Statistical Physics

Specific heat anomalies of small quantum systems subjected to finite baths

Hideo Hasegawa

J. Math. Phys. 52, 123301 (2011); http://dx.doi.org/10.1063/1.3669485 (18 pages)

Online Publication Date: 20 December 2011

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We have studied the specific heat of the (NS + NB) model for an NS-body harmonic oscillator (HO) system which is strongly coupled to an NB-body HO bath without dissipation. The system specific heat of CS(T) becomes NSkB at T → ∞ and vanishes at T = 0 in accordance with the third law of thermodynamics. The calculated CS(T) at low temperatures is not proportional to NS and shows an anomalous temperature dependence, strongly depending on NS, NB, and the system-bath coupling. In particular at very low (but finite) temperatures, it may become negative for a strong system-bath coupling, which is in contrast with non-negative specific heat of a HO system with NS = 1 reported by Ingold, Hänggi, and Talkner [Phys. Rev. E 79, 061105 (2005)]. Our calculation indicates an importance of taking account of finite NS in studying open quantum systems which may include an arbitrary number of particles in general.
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05.30.-d Quantum statistical mechanics
03.65.Ge Solutions of wave equations: bound states
03.65.Yz Decoherence; open systems; quantum statistical methods
back to top Methods of Mathematical Physics

Existence of ground states of hydrogen-like atoms in relativistic quantum electrodynamics. II. The no-pair operator

Martin Könenberg, Oliver Matte, and Edgardo Stockmeyer

J. Math. Phys. 52, 123501 (2011); http://dx.doi.org/10.1063/1.3658863 (34 pages)

Online Publication Date: 1 December 2011

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We consider a hydrogen-like atom in a quantized electromagnetic field which is modeled by means of a no-pair operator acting in the positive spectral subspace of the free Dirac operator minimally coupled to the quantized vector potential. We prove that the infimum of the spectrum of the no-pair operator is an evenly degenerate eigenvalue. In particular, we show that the bottom of its spectrum is strictly less than its ionization threshold. These results hold true, for arbitrary values of the fine-structure constant and the ultraviolet cut-off and for all Coulomb coupling constants less than the critical one of the Brown-Ravenhall model, 2/(2/π + π/2). For Coulomb coupling constants larger than the critical one, we show that the quadratic form of the no-pair operator is unbounded below. Along the way we discuss the domains and operator cores of the semi-relativistic Pauli-Fierz and no-pair operators, for Coulomb coupling constants less than or equal to the critical ones.
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31.30.jc Relativistic corrections to atomic structure and properties
12.20.Ds Specific calculations
31.50.Bc Potential energy surfaces for ground electronic states

Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras

Joakim Arnlind, Abdenacer Makhlouf, and Sergei Silvestrov

J. Math. Phys. 52, 123502 (2011); http://dx.doi.org/10.1063/1.3653197 (13 pages)

Online Publication Date: 5 December 2011

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As n-ary operations, generalizing Lie and Poisson algebras, arise in many different physical contexts, it is interesting to study general ways of constructing explicit realizations of such multilinear structures. Generically, they describe the dynamics of a physical system, and there is a need of understanding their quantization. Hom-Nambu-Lie algebras provide a framework that might be an appropriate setting in which n-Lie algebras (n-ary Nambu-Lie algebras) can be deformed, and their quantization studied. We present a procedure to construct (n + 1)-ary Hom-Nambu-Lie algebras from n-ary Hom-Nambu-Lie algebras equipped with a generalized trace function. It turns out that the implications of the compatibility conditions, that are necessary for this construction, can be understood in terms of the kernel of the trace function and the range of the twisting maps. Furthermore, we investigate the possibility of defining (n + k)-Lie algebras from n-Lie algebras and a k-form satisfying certain conditions.
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02.20.Sv Lie algebras of Lie groups

Split 3-Lie algebras

Antonio J. Calderón Martín and M. Forero Piulestán

J. Math. Phys. 52, 123503 (2011); http://dx.doi.org/10.1063/1.3664752 (16 pages)

Online Publication Date: 6 December 2011

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In order to begin an approach to the structure of 3-Lie algebras (with restrictions neither on the dimension nor on the base field), we introduce the class of split 3-Lie algebras as the natural extension of the class of split Lie algebras. By developing techniques of connections of roots for this kind of ternary algebras, we show that any of such split 3-Lie algebras math is of the form math = U+∑jIj with U a subspace of the 0-root space math0 and any Ij a well described ideal of math, satisfying [Ij,math,Ik] = 0 if j ≠ k. Under certain conditions the simplicity of math is characterized and it is shown that math is the direct sum of the family of its minimal ideals, each one being a simple split 3-Lie algebra.
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02.20.Sv Lie algebras of Lie groups

Constraint rescaling in refined algebraic quantisation: Momentum constraint

Jorma Louko and Eric Martínez-Pascual

J. Math. Phys. 52, 123504 (2011); http://dx.doi.org/10.1063/1.3664336 (16 pages)

Online Publication Date: 7 December 2011

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We investigate refined algebraic quantisation within a family of classically equivalent constrained Hamiltonian systems that are related to each other by rescaling a momentum-type constraint. The quantum constraint is implemented by a rigging map that is motivated by group averaging but has a resolution finer than what can be peeled off from the formally divergent contributions to the averaging integral. Three cases emerge, depending on the asymptotics of the rescaling function: (i) quantisation is equivalent to that with identity scaling; (ii) quantisation fails, owing to nonexistence of self-adjoint extensions of the constraint operator; (iii) a quantisation ambiguity arises from the self-adjoint extension of the constraint operator, and the resolution of this purely quantum mechanical ambiguity determines the superselection structure of the physical Hilbert space. Prospects of generalising the analysis to systems with several constraints are discussed.
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03.65.Fd Algebraic methods
02.10.-v Logic, set theory, and algebra

Holomorphic supercurves and supersymmetric sigma models

Josua Groeger

J. Math. Phys. 52, 123505 (2011); http://dx.doi.org/10.1063/1.3665710 (21 pages)

Online Publication Date: 9 December 2011

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We introduce a natural generalisation of holomorphic curves to morphisms of supermanifolds, referred to as holomorphic supercurves. More precisely, supercurves are morphisms from a Riemann surface, endowed with the structure of a supermanifold which is induced by a holomorphic line bundle, to an ordinary almost complex manifold. They are called holomorphic if a generalised Cauchy-Riemann condition is satisfied. We show, by means of an action identity, that holomorphic supercurves are special extrema of a supersymmetric action functional.
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11.10.Lm Nonlinear or nonlocal theories and models
11.30.Pb Supersymmetry
11.10.Cd Axiomatic approach

Uniqueness of the solution to inverse scattering problem with scattering data at a fixed direction of the incident wave

A. G. Ramm

J. Math. Phys. 52, 123506 (2011); http://dx.doi.org/10.1063/1.3666985 (12 pages)

Online Publication Date: 9 December 2011

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Let q(x) be real-valued compactly supported sufficiently smooth function. It is proved that the scattering data A(β, α0, k) ∀β ∈ S2, ∀k > 0, determine q uniquely. Here, α0 ∈ S2 is a fixed direction of the incident plane wave.
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03.65.Ge Solutions of wave equations: bound states
02.30.Zz Inverse problems

Construction of basis functions with crystal symmetry for the spin-cluster expansion of the magnetic energy on the atomic scale

F. Dietermann, R. Singer, and M. Fähnle

J. Math. Phys. 52, 123507 (2011); http://dx.doi.org/10.1063/1.3668054 (5 pages)

Online Publication Date: 12 December 2011

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For a crystal with symmetry according to a space-group G, a set of linearly independent and orthogonal basis functions for the spin-cluster expansion of the magnetic energy on the atomic scale is constructed, which is invariant under the action of each g∈G. This basis set is found by linear combinations of the basis functions invariant under global rotations of the magnetic moments, as defined in a former paper [R. Singer and M. Fähnle, J. Math. Phys. 47, 113503 (2006)10.1063/1.2364181].
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61.50.Ah Theory of crystal structure, crystal symmetry; calculations and modeling
05.70.Ce Thermodynamic functions and equations of state
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