Volume 43, Issue 8, August 2002
Index of content:
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Symmetry breaking regime in the nonlinear Hartree equation
View Description Hide DescriptionThe present article is concerned with minimizers of the attractive Hartree energy functional on for a general class of external potentials and twobody interactions of positive type, with We prove spontaneous symmetry breaking in the large coupling limit. A numerical investigation visualizes this regime in the example of an external double well potential.

Bound states in straight quantum waveguides with combined boundary conditions
View Description Hide DescriptionWe investigate the discrete spectrum of the Hamiltonian describing a quantum particle living in the twodimensional straight strip. We impose the combined Dirichlet and Neumann boundary conditions on different parts of the boundary. Several statements on the existence or the absence of the discrete spectrum are compared for two models with combined boundary conditions. Examples of eigenfunctions and eigenvalues are computed numerically.

Bogoliubov transformations and exact isolated solutions for simple nonadiabatic Hamiltonians
View Description Hide DescriptionWe present a new method for finding isolated exact solutions of a class of nonadiabatic Hamiltonians of relevance to quantum optics and allied areas. Central to our approach is the use of Bogoliubov transformations of the bosonic fields in the models. We demonstrate the simplicity and efficiency of this method by applying it to the Rabi Hamiltonian.

The Dirac operator in a fermion bag background in dimensions and generalized supersymmetric quantum mechanics
View Description Hide DescriptionWe show that the spectral theory of the Dirac operator in a static background in space–time dimensions, is underlined by a certain novel generalization of supersymmetric quantum mechanics, and we explore some of its mathematical and physical consequences.

Stabilization of impurity states in crossed magnetic and electric fields
View Description Hide DescriptionIt is shown that the renormalizability of the zerorange interaction in the twodimensional space is always followed by the existence of a bound state, which is not true for odddimensional spaces. A renormalization procedure is defined and the exact retarded Green’s function for electrons moving in two dimensions and interacting with both crossed magnetic and electric fields and an attractive zerorange interaction is constructed. Imaginary parts of poles of this Green’s function determine lifetimes of quasibound (resonance) states. It is shown that for some particular parameters the stabilization against decay occurs even for strong electric fields.

PseudoHermiticity versus symmetry III: Equivalence of pseudoHermiticity and the presence of antilinear symmetries
View Description Hide DescriptionWe show that a diagonalizable (nonHermitian) Hamiltonian is pseudoHermitian if and only if it has an antilinear symmetry, i.e., a symmetry generated by an invertible antilinear operator. This implies that the eigenvalues of are real or come in complex conjugate pairs if and only if possesses such a symmetry. In particular, the reality of the spectrum of implies the presence of an antilinear symmetry. We further show that the spectrum of is real if and only if there is a positivedefinite innerproduct on the Hilbert space with respect to which is Hermitian or alternatively there is a pseudocanonical transformation of the Hilbert space that maps into a Hermitian operator.

Energy levels and wave functions of vector bosons in a homogeneous magnetic field
View Description Hide DescriptionWe aimed to obtain the energy levels of massive spin1 particles moving in a constant magnetic field. The method used here is completely algebraic. In the process to obtain the energy levels the wave function is expressed in terms of Laguerre polynomials.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


The propagator of the radial Dirac equation
View Description Hide DescriptionThe propagator for the radial Dirac equation is explicitly constructed. It turns out to be a distribution of order zero, but it is shown that there exists no pathspace measure associated with this equation.

 GENERAL RELATIVITY AND GRAVITATION


Properties of the symplectic structure of general relativity for spatially bounded space–time regions
View Description Hide DescriptionWe continue a previous analysis of the covariant Hamiltonian symplectic structure of general relativity for spatially bounded regions of space–time. To allow for wide generality, the Hamiltonian is formulated using any fixed hypersurface, with a boundary given by a closed spacelike twosurface. A main result is that we obtain Hamiltonians associated with Dirichlet and Neumann boundary conditions on the gravitational field coupled to matter sources, in particular a Klein–Gordon field, an electromagnetic field, and a set of Yang–Mills–Higgs fields. The Hamiltonians are given by a covariant form of the Arnowitt–Deser–Misner (ADM) Hamiltonian modified by a surface integral term that depends on the particular boundary conditions. The general form of this surface integral involves an underlying “energymomentum” vector in the space–time tangent space at the spatial boundary twosurface. We give examples of the resulting Dirichlet and Neumann vectors for topologically spherical twosurfaces in Minkowski space–time, spherically symmetric space–times, and stationary axisymmetric space–times. Moreover, we establish the relation between these vectors and the ADM energymomentum vector for a twosurface taken in a limit to be spatial infinity in asymptotically flat space–times. We also discuss the geometrical properties of the Dirichlet and Neumann vectors and obtain several striking results relating these vectors to the mean curvature and normal curvature connection of the twosurface. Most significantly, the part of the Dirichlet vector normal to the twosurface depends only on the space–time metric at this surface and thereby defines a geometrical normal vector field on the twosurface. We show that this normal vector is orthogonal to the mean curvature vector, and its norm is the mean null extrinsic curvature, while its direction is such that there is zero expansion of the twosurface, i.e., the Lie derivative of the surface volume form in this direction vanishes. This leads to a direct relation between the Dirichlet vector and the condition for a spacelike twosurface to be (marginally) trapped.

On the problem of algebraic completeness for the invariants of the Riemann tensor. III.
View Description Hide DescriptionWe study the set CZ of invariants [Zakhary and Carminati, J. Math. Phys. 42, 1474 (2001)] for the class of space–times whose Ricci tensors possess a null eigenvector. We show that all cases are maximally backsolvable, in terms of sets of invariants from CZ, but that some cases are not completely backsolvable and these all possess an alignment between an eigenvector of the Ricci tensor with a repeated principal null vector of the Weyl tensor. We provide algebraically complete sets for each canonically different space–time and hence conclude with these results and those of a previous article [Carminati, Zakhary, and McLenaghan, J. Math. Phys. 43, 492 (2002)] that the CZ set is determining or maximal.

 DYNAMICAL SYSTEMS


Applications of Nambu mechanics to systems of hydrodynamical type
View Description Hide DescriptionWe show that the reduced Dubrovin–Novikov hydrodynamic type models are integrable Nambu mechanical systems admitting Lax triples.

A unified treatment of quartic invariants at fixed and arbitrary energy
View Description Hide DescriptionTwodimensional Hamiltonian systems admitting second invariants which are quartic in the momenta are investigated using the Jacobi geometrization of the dynamics. This approach allows for a unified treatment of invariants at both arbitrary and fixed energy. In the differential geometric picture, the quartic invariant corresponds to the existence of a fourth rank Killing tensor. Expressing the Jacobi metric in terms of a Kähler potential, the integrability condition for the existence of the Killing tensor at fixed energy is a nonlinear equation involving the Kähler potential. At arbitrary energy, further conditions must be imposed which lead to an overdetermined system with isolated solutions. We obtain several new integrable and superintegrable systems in addition to all previously known examples.

Multisymplectic geometry and multisymplectic Preissman scheme for the KP equation
View Description Hide DescriptionThe multisymplectic structure of the KP equation is obtained directly from the variational principal. Using the covariant De Donder–Weyl Hamilton function theories, we reformulate the KP equation to the multisymplectic form which was proposed by Bridges. From the multisymplectic equation, we can derive a multisymplectic numerical scheme of the KP equation which can be simplified to the multisymplectic 45 points scheme.

(2+1)dimensional component AKNS system: Painlevé integrability, infinitely many symmetries, similarity reductions and exact solutions
View Description Hide DescriptionThe AKNS system that is derived from the inner parameter dependent symmetry constraint of the KP equation is studied in detail. First, the Painlevé integrability of the model is proved by using the standard WTC and Kruskal approach. Using the formal series symmetry approach, the generalized KMV symmetry algebra and the related symmetry group are found. The twodimensional similarity partial differential equation reductions and the ordinary differential equation reductions are obtained from the generalized KMV symmetry algebra and the direct method. Abundant localized coherent structures are revealed by the variable separation approach. Some special types of the localized excitations like the multiple solitoffs, dromions, lumps, ring solitons, breathers and instantons are plotted also.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Commensurate harmonic oscillators: Classical symmetries
View Description Hide DescriptionThe symmetry properties of a classical dimensional harmonic oscillator with rational frequency ratios are studied from a global point of view. A commensurateoscillator possesses the same number of globally defined constants of motion as an isotropicoscillator. In both cases invariant phasespace functions form the algebra with respect to the Poisson bracket. In the isotropic case, the phasespace flows generated by the invariants can be integrated globally to a set of finite transformations isomorphic to the group For a commensurate oscillator, however, the group of symmetry transformations is found to exist only on a reduced phase space, due to unavoidable singularities of the flow in the full phase space. It is therefore crucial to distinguish carefully between local and global definitions of symmetry transformations in phase space. This result solves the longstanding problem of which symmetry to associate with a commensurate harmonic oscillator.

On a low energy bound in a class of chiral field theories with solitons
View Description Hide DescriptionA low energy bound for static classical solutions in a class of chiral solitonic field theories related to the infrared physics of the Yang–Mills theory is established.

 STATISTICAL PHYSICS


The XXZ spin chain at Bethe roots, symmetric functions, and determinants
View Description Hide DescriptionA number of conjectures have been given recently concerning the connection between the antiferromagnetic XXZ spin chain at and various symmetry classes of alternating sign matrices. Here we use the integrability of the XXZ chain to gain further insight into these developments. In doing so we obtain a number of new results using Baxter’s function for the XXZ chain for periodic, twisted and open boundary conditions. These include expressions for the elementary symmetric functions evaluated at the ground state solution of the Bethe roots. In this approach Schur functions play a central role and enable us to derive determinant expressions which appear in certain natural double products over the Bethe roots. When evaluated these give rise to the numbers counting different symmetry classes of alternating sign matrices.

Chiral mixtures
View Description Hide DescriptionAn index evaluating the amount of chirality of a mixture of colored random variables is defined. Properties are established. Extreme chiral mixtures are characterized and examples are given. Connections between chirality, Wasserstein distances, and least squares Procrustes methods are pointed out.

 METHODS OF MATHEMATICAL PHYSICS


Twisted duality of the CARalgebra
View Description Hide DescriptionWe give a complete proof of the twisted duality property of the (selfdual) CARAlgebra in any Fock representation. The proof is based on the natural Halmos decomposition of the (reference) Hilbert space when two suitable closed subspaces have been distinguished. We use modular theory and techniques developed by Kato concerning pairs of projections in some essential steps of the proof. As a byproduct of the proof we obtain an explicit and simple formula for the graph of the modular operator. This formula can be also applied to fermionic free nets, hence giving a formula of the modular operator for any double cone.

Fusion bases as facets of polytopes
View Description Hide DescriptionA new way of constructing fusion bases (i.e., the set of inequalities governing fusion rules) out of fusion elementary couplings is presented. It relies on a polytope reinterpretation of the problem: the elementary couplings are associated with the vertices of the polytope while the inequalities defining the fusion basis are the facets. The symmetry group of the polytope associated with the lowest rank affine Lie algebras is found; it has order 24 for 432 for and quite surprisingly, it reduces to 36 for while it is only of order 4 for This drastic reduction in the order of the symmetry group as the algebra gets more complicated is rooted in the presence of many linear relations between the elementary couplings that break most of the potential symmetries. For and it is shown that the fusionbasis defining inequalities can be generated from few (one and two, respectively) elementary ones. For new symmetries of the fusion coefficients are found.
