Volume 45, Issue 9, September 2004
Index of content:
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Analytic representations based on coherent states and Robertson intelligent states
View Description Hide DescriptionRobertson intelligent states which minimize the Schrödinger–Robertson uncertainty relation are constructed as eigenstates of a linear combination of Weyl generators of the algebra. The construction is based on the analytic representations of coherent states. New classes of coherent and squeezed states are explicitly derived.

 METHODS OF MATHEMATICAL PHYSICS


Solutions for a fractional nonlinear diffusion equation: Spatial time dependent diffusion coefficient and external forces
View Description Hide DescriptionWe analyze a generalized diffusion equation which extends some known equations such as the fractional diffusion equation and the porous medium equation. We start our investigation by considering the linear case and the nonlinear case afterward. The linear case is discussed taking fractional time and spatial derivatives into account in a unified approach. We also discuss the modifications that emerge by employing simple drifts and the diffusion coefficient given by For the nonlinear case, we study scaling behavior of the time in connection with the asymptotic behavior for the solution of the nonlinear fractional diffusion equation.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


High energy asymptotics of the magnetic spectral shift function
View Description Hide DescriptionWe consider the threedimensional Schrödinger operators and where , is a magnetic potential generating a constant magnetic field of strength , and where satisfies certain regularity conditions. Then the spectral shift function for the pair of operators , is welldefined for energies , . We study the asymptotic behavior of as , , , where . We obtain a Weyltype formula .

 DYNAMICAL SYSTEMS


Global periodic attractor for strongly damped wave equations with timeperiodic driving force
View Description Hide DescriptionIn this paper, we consider the existence of a global periodic attractor for a strongly damped nonlinear waveequation with timeperiodic driving force under homogeneous Dirichlet boundary condition. It is proved that in certain parameter region, for arbitrary timeperiodic driving force, the system has a unique periodic solution attracting any bounded set exponentially. This implies that the system behaves exactly as a onedimensional system. We mention, in particular, that the obtained result can be used to prove the existence of global periodic attractor of the usual damped and driven wave equations.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Energy momentum, wave velocities and characteristic shocks in Euler’s variational equations with application to the Born–Infeld theory
View Description Hide DescriptionWe consider the Euler’s variational equations deriving from a general Lagrangian. Under the assumption of convexity of energy, we write down some inequalities for the energymomentum tensor including Hawking–Ellis energy conditions. We show that there exists the same number of positive and negative wave velocities and no velocity can change sign. Finally, we study the structure of the characteristic shocks with particular attention to the generalized Born–Infeld Lagrangian describing the electron with spin.

 METHODS OF MATHEMATICAL PHYSICS


On the initial boundary value problem for a shallow water equation
View Description Hide DescriptionIn this paper, we obtain the existence and uniqueness of the local strong solutions to the initial boundary problem for a onedimensional shallowwater equation (Camassa–Holm equation) on the halfspace with initial data The solution is obtained as a limit of the solutions for a class of approximation problems. We also establish the global result of the corresponding solution, provided that the initial data satisfies certain positivity condition.

 GENERAL RELATIVITY AND GRAVITATION


Asymptotic analysis of field commutators for Einstein–Rosen gravitational waves
View Description Hide DescriptionWe give a detailed study of the asymptotic behavior of field commutators for linearly polarized, cylindrically symmetric gravitational waves in different physically relevant regimes. We also discuss the necessary mathematical tools to carry out our analysis. Field commutators are used here to analyze microcausality, in particular the smearing of light cones owing to quantum effects. We discuss in detail several issues related to the semiclassical limit of quantum gravity, in the simplified setting of the cylindrical symmetry reduction considered here. We show, for example, that the small G behavior is not uniform in the sense that its functional form depends on the causal relationship between space–time points. We consider several physical issues relevant for this type of models such as the emergence of large gravitational effects.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Integral representation of onedimensional three particle scattering for δ function interactions
View Description Hide DescriptionThe Schrödinger equation, in hyperspherical coordinates, is solved in closed form for a system of three particles on a line, interacting via pair delta functions. This is for the case of equal masses and potential strengths. The interactions are replaced by appropriate boundary conditions. This leads then to requiring the solution of a freeparticle Schrödinger equation subject to these boundary conditions. A generalized Kontorovich–Lebedev transformation is used to write this solution as an integral involving a product of Bessel functions and pseudoSturmian functions. The coefficient of the product is obtained from a threeterm recurrence relation, derived from the boundary condition. The contours of the Kontorovich–Lebedev representation are fixed by the asymptotic conditions. The scattering matrix is then derived from the exact solution of the recurrence relation. The wavefunctions that are obtained are shown to be equivalent to those derived by McGuire. The method can clearly be applied to a larger number of particles and hopefully might be useful for unequal masses and potentials.

 GENERAL RELATIVITY AND GRAVITATION


On the general structure of Ricci collineations for type B warped space–times
View Description Hide DescriptionA complete study of the structure of Ricci collineations for type B warped space–times is carried out. This study can be used as a method to obtain these symmetries in such space–times. Special cases as reducible space–times, and plane and spherical symmetric space–times are considered specifically.

 DYNAMICAL SYSTEMS


On a theorem by Treves
View Description Hide DescriptionAccording to a theorem by Treves [Duke Math. J.DUMJAO 108, 251 (2001)], the conserved functionals of the equation vanish on each formal Laurent series . We propose a new, very simple geometrical proof for this statement.

 STATISTICAL PHYSICS


The canonical perfect Bose gas in Casimir boxes
View Description Hide DescriptionWe study the problem of Bose–Einstein condensation in the perfect Bose gas in the canonical ensemble, in anisotropically dilated rectangular parallelepipeds (Casimir boxes). We prove that in the canonical ensemble for these anisotropic boxes there is the same type of generalized Bose–Einstein condensation as in the grandcanonical ensemble for the equivalent geometry. However the amount of condensate in the individual states is different in some cases and so are the fluctuations.

 METHODS OF MATHEMATICAL PHYSICS


Evaluation of phasemodulated lattice sums
View Description Hide DescriptionAn exact evaluation of twodimensional phasemodulated lattice sums of the form is presented in terms of the Jacobian theta functions. The result generalizes the identity derived by M. L. Glasser [J. Math. Phys. 15, 188 (1974)] to allow for evaluation on nonrectangular lattices. The generalized identity is also applied to a problem in vortex dynamics.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Reduction of the classical MICZKepler problem to a twodimensional linear isotropic harmonic oscillator
View Description Hide DescriptionThe classical MICZKepler problem is shown to be reducible to an isotropic twodimensional system of linear harmonic oscillators and a conservation law in terms of new variables related to the Ermanno–Bernoulli constants and the components of the Poincaré vector. An algorithmic route to linearization is shown based on Lie symmetry analysis and the reduction method [ Nucci, J. Math. Phys. 37, 1772 (1996) ]. First integrals are also obtained by symmetry analysis and the reduction method [ Marcelli and Nucci,J. Math. Phys. 44, 2111 (2002) ].

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


The existence problem for dynamics of dissipative systems in quantum probability
View Description Hide DescriptionMotivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following algebraic setting: A given Hermitian dissipative mapping is densely defined in a unital algebra . The identity element in is also in the domain of . Completely dissipative maps are defined by the requirement that the induced maps, , are dissipative on the by complex matrices over for all . We establish the existence of different types of maximal extensions of completely dissipative maps. If the enveloping von Neumann algebra of is injective, we show the existence of an extension of which is the infinitesimal generator of a quantum dynamical semigroup of completely positive maps in the von Neumann algebra. If is a given wellbehaved *derivation, then we show that each of the maps is completely dissipative.

Extremal covariant quantum operations and positive operator valued measures
View Description Hide DescriptionWe consider the convex sets of QO’s (quantum operations) and POVM’s (positive operator valued measures) which are covariant under a general finitedimensional unitary representation of a group. We derive necessary and sufficient conditions for extremality, and give general bounds for ranks of the extremal POVM’s and QO’s. Results are illustrated on the basis of simple examples.

 METHODS OF MATHEMATICAL PHYSICS


Derivation of the supersymmetric HarishChandra integral for
View Description Hide DescriptionThe previous supersymmetric generalization of the unitary HarishChandra integral prompted the conjecture that the HarishChandra formula should have an extension to superspaces. We prove this conjecture for the unitary orthosymplectic supermanifold . To this end, we construct and solve an eigenvalueequation.

 STATISTICAL PHYSICS


An Ising model with three competing interactions on a Cayley tree
View Description Hide DescriptionIn this paper we consider an Ising model with three competing restricted interactions on the Cayley tree . The translation invariant and periodic Gibbs measures for these models are investigated and the problem of the phase transition in these classes is solved.

 METHODS OF MATHEMATICAL PHYSICS


Finite size universe or perfect squash problem
View Description Hide DescriptionWe give a physical notion to all selfadjoint extensions of the operator in the finite interval. It appears that these extensions realize different nonunitary equivalent representations of CCR and are related to the momentum operator viewed from different inertial systems. This leads to the generalization of Galilei equivalence principle and gives a new insight into the quantum correspondence rule. It is possible to get transformation laws of the wave function under Galilei transformation for any scalar potential. This generalizes the mass superselection rule. There is also given a new and general interpretation of a momentum representation of the wave function. It appears that consistent treatment of this problem leads to the timedependent interactions and to the abrupt switchingoff of the interaction.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Semiclassical limit for multistate Klein–Gordon systems: almost invariant subspaces, and scattering theory
View Description Hide DescriptionBy using the method of Helffer and Sjöstrand to construct Moyal projections, we extend the almost invariant subspacetheory to the semiclassical context. Applications to the semiclassical limit for two component Klein–Gordon Hamiltonian are given. More precisely, under the conditions that the potential is analytic and its eigenvalues never cross we prove that the scattering matrix is block diagonal up to exponentially small errors. Also, we show how the existence of almost invariant subspaces leads to the existence of quasimodes with exponentially long lifetimes.
