Volume 51, Issue 3, March 2010
Index of content:
 ARTICLES


Quantum Mechanics (General and Nonrelativistic)

fold supersymmetry and quasisolvability associated with Laguerre polynomials
View Description Hide DescriptionWe construct a new family of quasisolvable and fold supersymmetric quantum systems where each Hamiltonian preserves an exceptional polynomialsubspace of codimension 2. We show that the family includes as a particular case the recently reported rational radial oscillator potential whose eigenfunctions are expressed in terms of the Laguerre polynomials of the second kind. In addition, we find that the two kinds of the Laguerre polynomials are ingeniously connected with each other by the fold supercharge.

Well posedness for the nonlinear Klein–Gordon–Schrödinger equations with heterointeractions
View Description Hide DescriptionIn this paper we study the well posedness of solution to a certain system of nonlinear Klein–Gordon–Schrödinger equations in three space dimensions. Basing on the Strichartz estimates, we obtain the global existence, uniqueness of the solutions, and continuous dependence with respect to initial data in the Sobolev spaces of low regularities by setting appropriate contraction and taking difference estimates.

Scattering theory for twobody quantum systems with singular potentials in a timeperiodic electric field^{a)}
View Description Hide DescriptionRecently, the first author [Adachi, “Asymptotic completeness for body quantum systems with longrange interactions in a timeperiodic electric field,” Commun. Math. Phys.275, 443 (2007)] proved the asymptotic completeness for body quantum systems with longrange interactions in a timeperiodic electric field whose mean in time is nonzero by obtaining propagation estimates for the physical propagator. However, in his work, it is needed that potentials under consideration are sufficiently smooth. In this paper, when , we prove the asymptotic completeness of (modified) wave operators under the assumption that the potential has the local singularity of type when and , where is the space dimension. We also discuss the modifiers in the position representation, which are used in the definition of the modified wave operators in the longrange case.

The quantum body problem and the auxiliary field method
View Description Hide DescriptionApproximate analytical energy formulas for body semirelativistic Hamiltonians with one and twobody interactions are obtained within the framework of the auxiliary field method. This method has already been proven to be a powerful technique in the case of twobody problems. A general procedure is given and applied to various Hamiltonians of interest, in atomic and hadronic physics in particular. A test of formulas is performed for baryons described as a threequark system.

The Liouville equation for singular ergodic magnetic Schrödinger operators
View Description Hide DescriptionWe study the time evolution of a density matrix in a quantum mechanical system described by an ergodic magnetic Schrödinger operator with singular magnetic and electric potentials, the electric field being introduced adiabatically. We construct a unitary propagator that solves weakly the corresponding timedependent Schrödinger equation and solve a Liouville equation in an appropriate Hilbert space.

Quantum Information and Computation

Structural approach to unambiguous discrimination of two mixed quantum states
View Description Hide DescriptionWe analyze the optimal unambiguous discrimination of two arbitrary mixed quantum states. We show that the optimal measurement is unique and we present this optimal measurement for the case where the rank of the density operator of one of the states is at most 2 (“solution in four dimensions”). The solution is illustrated by some examples. The optimality conditions proven by Eldar et al. [Phys. Rev. A69, 062318 (2004)] are simplified to an operational form. As an application we present optimality conditions for the measurement, when only one of the two states is detected. The current status of optimal unambiguous state discrimination is summarized via a general strategy.

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

Path integral solutions for Klein–Gordon particle in vector plus scalar generalized Hulthén and Woods–Saxon potentials
View Description Hide DescriptionThe Green’s function for a Klein–Gordon particle under the action of vector plus scalar deformed Hulthén and Woods–Saxon potentials is evaluated by exact path integration. Explicit path integration leads to the Green’s function for different shapes of the potentials. From the singularities of the latter Green’s function, the bound states are extracted. For and , the analytic expression of the energy spectrum and the normalized wave functions for the states are obtained within the framework of an approximation to the centrifugal term. When the deformation parameter is or , it is found that the quantization conditions are transcendental equations involving the hypergeometric function that require a numerical solution for the state energy levels. Particular cases of these potentials are also discussed briefly.

Quantum fields on star graphs with bound states at the vertex
View Description Hide DescriptionWe investigate the propagation of a massless scalar field on a star graph, modeling the junction of quantum wires. The vertex of the graph is represented by a pointlike impurity (defect), characterized by a onebody scattering matrix. The general case of offcritical scattering matrix with bound and/or antibound states is considered. We demonstrate that the contribution of these states to the scalar field is fixed by causality (local commutativity), which is the key point of our investigation. Two different regimes of the theory emerge at this stage. If bound sates are absent, the energy is conserved and the theory admits unitary time evolution. The behavior changes if bound states are present because each such state generates a kind of damped harmonic oscillator in the spectrum of the field. These oscillators lead to the breakdown of timetranslation invariance. In both regimes we investigate in this framework the electromagnetic conductance of the Luttinger liquid on the quantum wire junction. We derive an explicit expression for the conductance in terms of the scattering matrix and show that antibound and bound states have a different impact, giving rise to oscillations with exponentially damped and growing amplitudes, respectively.

Bounds on diatomic molecules in a relativistic model
View Description Hide DescriptionWe consider diatomic systems in which the kinetic energy of the electrons is treated in a simple relativistic model. The Born–Oppenheimer approximation is assumed. We investigate questions of stability, deducing bounds on the number of electrons, the binding energy, and the equilibrium bond distance . We use a known localization argument adopted to the present relativistic setting, with particular consideration of the critical point of stability, as well as the recently proved relativistic Scott correction.

Phaseintegral solution of the radial Dirac equation
View Description Hide DescriptionA phaseintegral (WKB) solution of the radial Dirac equation is constructed, retaining perfect symmetry between the two components of the wave function and introducing no singularities except at the classical transition points. The potential is allowed to be the time component of a fourvector, a Lorentz scalar, a pseudoscalar, or any combination of these. The key point in the construction is the transformation from two coupled firstorder equations constituting the radial Dirac equation to a single secondorder Schrödingertype equation. This transformation can be carried out in infinitely many ways, giving rise to different secondorder equations but with the same spectrum. A unique transformation is found that produces a particularly simple secondorder equation and correspondingly simple and wellbehaved phaseintegral solutions. The resulting phaseintegral formulas are applied to unbound and bound states of the Coulomb potential. For bound states, the exact energy levels are reproduced.

On the renormalization of the complex scalar free field theory
View Description Hide DescriptionPolar coordinates are used for the complex scalar free field in dimensions. The resulting nonrenormalizable theory is healed by using a recently proposed symmetric subtraction procedure. The existence of the coordinate transformation is proved by construction.

General Relativity and Gravitation

Existence and Newtonian limit of nonlinear bound states in the Einstein–Dirac system
View Description Hide DescriptionAn analysis is given of particlelike nonlinear bound states in the Newtonian limit of the coupled Einstein–Dirac system introduced by Finster et al. [“Particlelike solutions of the EinsteinDiracMaxwell equations,” Phys. Lett. A259, 431 (1999)]. A proof is given of the existence of these bound states in the almost Newtonian regime, and it is proven that they may be approximated by the energy minimizing solution of the Newton–Schrödinger system obtained by Lieb.

Involutivity of field equations
View Description Hide DescriptionWe prove involutivity of Einstein, Einstein–Maxwell, and other field equations by calculating the Spencer cohomology of these systems. The relation with Cartan method is traced in detail. Basic implications through Cartan–Kähler theory are derived.

Conformal linear gravity in de Sitter space
View Description Hide DescriptionIt has been shown that the theory of linear conformal quantum gravity must include a tensor field of rank3 and mixed symmetry [Binegar et al., Phys. Rev. D27, 2249 (1983)]. In this paper, we obtain the corresponding field equation in de Sitter space. Then, in order to relate this field with the symmetric tensor field of rank2, related to graviton, we will define homomorphisms between them. Our main result is that if one insists to be a unitary irreducible representation of de Sitter and conformal groups, it must satisfy a field equation of order of 6, which is obtained.

Static spherically symmetric solutions of the SO(5) Einstein Yang–Mills equations
View Description Hide DescriptionGlobally regular (i.e., asymptotically flat and regular interior) spherically symmetric and localized (“particlelike”) solutions of the coupled Einstein Yang–Mills (EYM) equations with gauge group SU(2) have been known for more than 20 years, yet their properties are still not well understood. Spherically symmetric Yang–Mills fields are classified by a choice of isotropy generator and SO(5) is distinguished as the simplest gauge group having a model with a nonAbelian residual (little) group, , which admits globally regular particlelike solutions. We exhibit an algebraic gauge condition which normalizes the residual gauge freedom to a finite number of discrete symmetries. This generalizes the wellknown reduction to the real magnetic potential in the original SU(2) YM model. Reformulating using gaugeinvariant polynomials dramatically simplifies the system and makes numerical search techniques feasible. We find three families of embedded SU(2) EYM equations within the SO(5) system, one of which was first detected only within the gaugeinvariant polynomial reduced system. Numerical solutions representing mixtures of the three SU(2) subsystems are found, classified by a pair of positive integers.

Dynamical Systems

Global attractors and their Hausdorff dimensions for a class of Kirchhoff models
View Description Hide DescriptionThis paper studies the existence, regularity, and Hausdorff dimensions of global attractors for a class of Kirchhoff models arising in elastoplastic flow. It proves that under rather mild conditions, the dynamical system associated with abovementioned models possesses in phase space a global attractor which has further regularity in and has finite Hausdorff dimension. For application, the fact shows that for the concerned elastoplastic flow the permanent regime (global attractor) can be observed when the excitation starts from any bounded set in phase space, and the dimension of the attractor, that is, the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow, is finite.

Random attractors for first order stochastic retarded lattice dynamical systems
View Description Hide DescriptionThis paper is concerned with the existence of random attractors for a general first order stochastic retarded lattice dynamical systems. It shows that, under suitable dissipative conditions, such a system possesses a random attractor which is a random compact invariant set. Furthermore, the ergodicity of the system is also proven.

Classical Mechanics and Classical Fields

Pendulum, elliptic functions, and relative cohomology classes
View Description Hide DescriptionRevisiting canonical integration of the classical pendulum around its unstable equilibrium, normal hyperbolic canonical coordinates are constructed and an identity between elliptic functions is found whose proof can be based on symplectic geometry and global relative cohomology. Alternatively it can be reduced to a well known identity between elliptic functions. Normal canonical actionangle variables are also constructed around the stable equilibrium and a corresponding identity is exhibited.

A geometric theory of thermal stresses
View Description Hide DescriptionIn this paper we formulate a geometric theory of thermal stresses. Given a temperature distribution, we associate a Riemannian materialmanifold to the body, with a metric that explicitly depends on the temperature distribution. A change in temperature corresponds to a change in the material metric. In this sense, a temperature change is a concrete example of the socalled referential evolutions. We also make a concrete connection between our geometric point of view and the multiplicative decomposition of deformation gradient into thermal and elastic parts. We study the stressfree temperature distributions of the finitedeformation theory using curvature tensor of the materialmanifold. We find the zerostress temperature distributions in nonlinear elasticity. Given an equilibrium configuration, we show that a change in the materialmanifold, i.e., a change in the material metric will change the equilibrium configuration. In the case of a temperature change, this means that given an equilibrium configuration for a given temperature distribution, a change in temperature will change the equilibrium configuration. We obtain the explicit form of the governing partial differential equations for this equilibrium change. We also show that geometric linearization of the present nonlinear theory leads to governing equations that are identical to those of the classical linear theory of thermal stresses.

Fluids

Burgers equation with a passive scalar: Dissipation anomaly and Colombeau calculus
View Description Hide DescriptionA connection between dissipation anomaly in fluid dynamics and Colombeau’s theory of products of distributions is exemplified by considering Burgers equation with a passive scalar. Besides the wellknown viscosityindependent dissipation of energy in the steadily propagating shock wave solution, the lesser known case of passive scalar subject to the shock wave is studied. An exact dependence of the dissipation rate of the passive scalar on the Prandtl number is given by a simple analysis: we show, in particular, for large . The passive scalar profile is shown to have a form of a sum of with suitably scaled , thereby implying the necessity to distinguish from when is large, where is the Heaviside function and is a positive integer. An incorrect result of would otherwise be obtained. This is a typical example where Colombeau calculus for products of weak solutions is required for a correct interpretation. A Cole–Hopftype transform is also given for the case of unit Prandtl number.
