Volume 52, Issue 12, December 2011
Index of content:
 ARTICLES


Quantum Mechanics (General and Nonrelativistic)

On the supremum and infimum of bounded quantum observables
View Description Hide DescriptionLet be the set of all bounded selfadjoint linear operators on a complex Hilbert space In 2006, Gudder [Math. Slovaca56, 573 (2006)] introduced a new order ≼ on Since then, the existence conditions and representations of the supremum and infimum of two elements in 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 and P is an orthogonal projection on 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.

Extreme covariant quantum observables in the case of an Abelian symmetry group and a transitive value space
View Description Hide DescriptionWe 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 Gspace. 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.

An alternative approach to Schrödinger equations with a spatially varying mass
View Description Hide DescriptionExtending 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 nonconstant masses.

Spectral problems for the Weylordered form of operators
View Description Hide DescriptionIn this paper, we consider quantization of powers of the ratio between the Hamiltonian coordinates for position and momentum in onedimensional systems. The domain of the operators consists of square integrable functions over a finite real interval to ensure boundedness and selfadjointness. The spectral problems for the operators that result from using Weylordering 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.

Exact results for finite quantum Hall systems of electrons at filling factor one: Disk geometry
View Description Hide DescriptionWe 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 manybody systems.

The propagator of the attractive deltaBose gas in one dimension
View Description Hide DescriptionWe consider the quantum δBose 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.

Prepotential approach to solvable rational extensions of Harmonic Oscillator and Morse potentials
View Description Hide DescriptionWe 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, DarbouxCrum, and DarbouxBäcklund transformations.

Relativistic and nonrelativistic bound states of the isotonic oscillator by NikiforovUvarov method
View Description Hide DescriptionA nonpolynomial onedimensional 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 E _{ n } and the wave functions ψ_{ n }(x) in terms of the associated Laguerre polynomials in the framework of the NikiforovUvarov method. Under the spin and pseudospin symmetric limits, the analytic eigenvalues and the corresponding twocomponent upper and lowerspinors of the Dirac particle are obtained in closed form.

Quantum Information and Computation

Existence of product vectors and their partial conjugates in a pair of spaces
View Description Hide DescriptionLet D and E be subspaces of the tensor product of the m and ndimensional complex spaces, with codimensions 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.

Threebythree bound entanglement with general unextendible product bases
View Description Hide DescriptionWe discuss the subject of unextendible product bases with the orthogonality condition dropped and we prove that the lowest rank nonseparable positivepartialtranspose 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 nonzero 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.

Description of rank four entangled states of two qutrits having positive partial transpose
View Description Hide DescriptionIt is known that some twoqutrit 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 fivedimensional subspaces of two qutrits which contain only finitely many product states (up to scalar multiple), and in particular those spanned by an UPB.

Classical and nonclassical randomness in quantum measurements
View Description Hide DescriptionThe space of positive operatorvalued probability measures on the Borel sets of a compact (or even locally compact) Hausdorff space X with values in , the algebra of linear operators acting on a ddimensional Hilbert space, is studied from the perspectives of classical and nonclassical convexity through a transform Γ that associates any positive operatorvalued measure ν with a certain completely positive linear map Γ(ν) of the homogeneous C*algebra into . This association is achieved by using an operatorvalued integral in which nonclassical random variables (that is, operatorvalued functions) are integrated with respect to positive operatorvalued 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 . The extremal and C*extremal points of are determined.

Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

The graviton propagator in de Donder gauge on de Sitter background
View Description Hide DescriptionWe 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.

General Relativity and Gravitation

Lorentzian covering space and homotopy classes
View Description Hide DescriptionWe 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.

Dynamical Systems

Dynamically defined measures and equilibrium states
View Description Hide DescriptionA 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.

Invariant algebraic surfaces for the reduced threewave interaction system
View Description Hide DescriptionIn this paper, we consider the threewave interaction system where γ, δ are real parameters. Our main results are the complete characterization of all values of the parameters γ, δ for which the threewave interaction system admits either invariant algebraic surfaces or algebraic (polynomial or rational) first integrals.

Classical Mechanics and Classical Fields

GoryachevChaplygin, Kovalevskaya, and BrdičkaEardleyNappiWitten ppwaves spacetimes with higher rank StäckelKilling tensors
View Description Hide DescriptionHidden symmetries of the GoryachevChaplygin and Kovalevskaya gyrostats spacetimes, as well as the BrdičkaEardleyNappiWitten ppwaves are studied. We find out that these spacetimes possess higher rank StäckelKilling tensors and that in the case of the ppwave spacetimes, the symmetry group of the StäckelKilling tensors is the wellknown NewtonHooke group.

Statistical Physics

Specific heat anomalies of small quantum systems subjected to finite baths
View Description Hide DescriptionWe have studied the specific heat of the (N S + N B ) model for an N S body harmonic oscillator (HO) system which is strongly coupled to an N B body HO bath without dissipation. The system specific heat of C S (T) becomes N S k B at T → ∞ and vanishes at T = 0 in accordance with the third law of thermodynamics. The calculated C S (T) at low temperatures is not proportional to N S and shows an anomalous temperature dependence, strongly depending on N S , N B , and the systembath coupling. In particular at very low (but finite) temperatures, it may become negative for a strong systembath coupling, which is in contrast with nonnegative specific heat of a HO system with N S = 1 reported by Ingold, Hänggi, and Talkner [Phys. Rev. E79, 061105 (2005)]. Our calculation indicates an importance of taking account of finite N S in studying open quantum systems which may include an arbitrary number of particles in general.

Methods of Mathematical Physics

Existence of ground states of hydrogenlike atoms in relativistic quantum electrodynamics. II. The nopair operator
View Description Hide DescriptionWe consider a hydrogenlike atom in a quantized electromagnetic field which is modeled by means of a nopair 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 nopair 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 finestructure constant and the ultraviolet cutoff and for all Coulomb coupling constants less than the critical one of the BrownRavenhall model, 2/(2/π + π/2). For Coulomb coupling constants larger than the critical one, we show that the quadratic form of the nopair operator is unbounded below. Along the way we discuss the domains and operator cores of the semirelativistic PauliFierz and nopair operators, for Coulomb coupling constants less than or equal to the critical ones.

Construction of nLie algebras and nary HomNambuLie algebras
View Description Hide DescriptionAs nary 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. HomNambuLie algebras provide a framework that might be an appropriate setting in which nLie algebras (nary NambuLie algebras) can be deformed, and their quantization studied. We present a procedure to construct (n + 1)ary HomNambuLie algebras from nary HomNambuLie 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 nLie algebras and a kform satisfying certain conditions.
