Volume 44, Issue 3, March 2003
Index of content:
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Critical energies in random palindrome models
View Description Hide DescriptionWe investigate the occurrence of critical energies—where the Lyapunov exponent vanishes—in random Schrödinger operators when the potentials have some local order, which we call random palindrome models. We give necessary and sufficient conditions for the presence of such critical energies: the commutativity of finite word elliptic transfer matrices. Finally, we perform some numerical calculations of the Lyapunov exponents showing their behavior near the critical energies and the respective time evolution of an initially localized wave packet, obtaining the exponent ruling the algebraic growth of the second momentum. We also consider special random palindrome models with oneletter bounded gap property; the transport effects of such long range order are showed numerically.

New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field
View Description Hide DescriptionThe perturbation series for the ground stateenergy of the hydrogen atom in the external magnetic field is summed via the sequence transformations. The formula for the largeorder behavior of the partial sums of the series is derived. From this formula a new general sequence transformation is suggested. This transformation contains free parameters that can be further optimized. It is shown that if the renormalization approach is used, the optimal choice of these parameters leads to the previously suggested Weniger transformation.

Preserving the measure of compatibility between quantum states
View Description Hide DescriptionIn this article after defining the abstract concept of compatibilitylike functions on quantum states, we prove that every bijective transformation on the set of all states which preserves such a function is implemented by an either unitary or antiunitary operator.

PseudoHermiticity and generalized PT and CPTsymmetries
View Description Hide DescriptionWe study certain linear and antilinear symmetry generators and involution operators associated with pseudoHermitian Hamiltonians and show that the theory of pseudoHermitian operators provides a simple explanation for the recent results of Bender, Brody and Jones (quantph/0208076) on the symmetry of a class of symmetric nonHermitian Hamiltonians. We present a natural extension of these results to the class of diagonalizable pseudoHermitian Hamiltonians with a discrete spectrum. In particular, we introduce generalized parity (P), timereversal (T), and chargeconjugation (C) operators and establish the PT and CPTinvariance of

Resolvent convergence of sphere interactions to point interactions
View Description Hide DescriptionWe consider the Hamiltonian which describes a sphere interaction We study the convergence of when in the norm resolvent sense. The existence of the zero resonance of affects the form of the limiting operator.

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


Gauge theories of Yang–Mills vector fields coupled to antisymmetric tensor fields
View Description Hide DescriptionA nonAbelian class of massless/massive nonlinear gauge theories of Yang–Mills vector potentials coupled to Freedman–Townsend antisymmetric tensor potentials is constructed in four space–time dimensions. These theories involve an extended Freedman–Townsendtype coupling between the vector and tensor fields, and a Chern–Simons mass term with the addition of a Higgstype coupling of the tensor fields to the vector fields in the massive case. Geometrical, field theoretic, and algebraic aspects of the theories are discussed in detail. In particular, the geometrical structure mixes and unifies features of Yang–Mills theory and Freedman–Townsend theory formulated in terms of Lie algebra valued curvatures and connections associated to the fields and nonlinear field strengths. The theories arise from a general determination of all possible geometrical nonlinear deformations of linear Abelian gauge theory for oneform fields and twoform fields with an Abelian Chern–Simons mass term in four dimensions. For this type of deformation (with typical assumptions on the allowed form considered for terms in the gauge symmetries and field equations), an explicit classification of deformation terms at firstorder is obtained, and uniqueness of deformation terms at all higher orders is proven. This leads to a uniqueness result for the nonAbelian class of theories constructed here.

Thermodynamic properties of the piece relativistic string
View Description Hide DescriptionThe thermodynamicfree energy is calculated for a gas consisting of the transverse oscillations of a piecewise uniform bosonic string. The string consists of parts of equal length, of alternating type I and type II material, and is relativistic in the sense that the velocity of sound everywhere equals the velocity of light. The present paper is a continuation of two earlier papers, one dealing with the Casimir energy of a piece string [I. Brevik and R. Sollie, J. Math. Phys. 38, 2774 (1997)], and another dealing with the thermodynamic properties of a string divided into two (unequal) parts [I. Brevik, A. A. Bytsenko, and H. B. Nielsen, Class. Quantum Grav. 15, 3383 (1998)]. Making use of the Meinardus theorem, we calculate the asymptotics of the level state density, and show that the critical temperatures in the individual parts are equal, for arbitrary space–time dimension If we find being the tension in part II. Thermodynamic interactions of parts related to high genus is also considered.

Adjoint operators, gauge invariant perturbations, and covariant symplectic form for black holes in string theory
View Description Hide DescriptionUsing a scheme of adjoint operators, we give a covariant and gauge invariant treatment for the perturbation theory of static charged black holes in string theory, valid for curvature below the Planck scale; conserved quantities and a covariant symplectic form on the phase space are explicitly constructed. Future extensions of the present results are discussed.

Spherically symmetric solutions of a boundary value problem for monopoles
View Description Hide DescriptionIn this article we study spherically symmetric monopoles, which are critical points for the Yang–Mills–Higgs functional over a disk in three dimensions, with prescribed degree and covariant constant at the boundary. This is a threedimensional gaugetheory generalization of the GinzburgLandau model in two dimensions.

Massive complex scalar field in the Kerr–Sen geometry: Exact solution of wave equation and Hawking radiation
View Description Hide DescriptionThe separated radial part of a massive complex scalar wave equation in the Kerr–Sen geometry is shown to satisfy the generalized spheroidal wave equation which is, in fact, a confluent Heun equation up to a multiplier. The Hawking evaporation of scalar particles in the Kerr–Sen black hole background is investigated by the Damour–Ruffini–Sannan method. It is shown that quantum thermal effect of the Kerr–Sen black hole has the same character as that of the Kerr–Newman black hole.

 GENERAL RELATIVITY AND GRAVITATION


Quasihomogeneous thermodynamics and black holes
View Description Hide DescriptionWe propose a generalized thermodynamics in which quasihomogeneity of the thermodynamic potentials plays a fundamental role. This thermodynamic formalism arises from a generalization of the approach presented in Ref. 1, and it is based on the requirement that quasihomogeneity is a nontrivial symmetry for the Pfaffian form It is shown that quasihomogeneous thermodynamics fits the thermodynamic features of at least some selfgravitating systems. We analyze how quasihomogeneous thermodynamics is suggested by black hole thermodynamics. Then, some existing results involving selfgravitating systems are also shortly discussed in the light of this thermodynamic framework. The consequences of the lack of extensivity are also recalled. We show that generalized Gibbs–Duhem equations arise as a consequence of quasihomogeneity of the thermodynamic potentials. An heuristic link between this generalized thermodynamic formalism and the thermodynamic limit is also discussed.

Leibnizian, Galilean and Newtonian structures of space–time
View Description Hide DescriptionThe following three geometrical structures on a manifold are studied in detail: Leibnizian: a nonvanishing oneform Ω plus a Riemannian metric 〈⋅,⋅〉 on its annhilator vector bundle. In particular, the possible dimensions of the automorphism group of a Leibnizian structure are characterized. Galilean: Leibnizian structure endowed with an affine connection ∇ (gauge field) which parallelizes Ω and 〈⋅,⋅〉. For any fixed vector field of observers an explicit Koszultype formula which reconstructs bijectively all the possible ∇’s from the gravitational and vorticity ω≔ rot fields (plus eventually the torsion) is provided. Newtonian: Galilean structure with 〈⋅,⋅〉 flat and a field of observers which is inertial (its flow preserves the Leibnizian structure and ω≡0). Classical concepts in Newtonian theory are revisited and discussed.

 DYNAMICAL SYSTEMS


Reciprocal transformations of Hamiltonian operators of hydrodynamic type: Nonlocal Hamiltonian formalism for linearly degenerate systems
View Description Hide DescriptionReciprocal transformations of Hamiltonian operators of hydrodynamic type are investigated. The transformed operators are generally nonlocal, possessing a number of remarkable algebraic and differentialgeometric properties. We apply our results to linearly degenerate semiHamiltonian systems in Riemann invariants, a typical example being Since all such systems are linearizable by appropriate (generalized) reciprocal transformations, our formulas provide an infinity of mutually compatible nonlocal Hamiltonian structures, explicitly parametrized by arbitrary functions of one variable.

Involution analysis of the partial differential equations characterizing Hamiltonian vector fields
View Description Hide DescriptionIn a recent article, certain underdetermined linear systems of partial differential equations connected with Lie–Poisson structures have been studied. They were constructed via power series solutions of the evolution equation for a given Hamiltonian. We extend the results to arbitrary Poissonmanifolds, correct an error in the case of degenerate Poisson structures, and show that these linear systems simply characterize Hamiltonian vector fields. Our basic tool is the formal theory of differential equations with its central concept of an involutive system.

Discrete symmetry’s chains and links between integrable equations
View Description Hide DescriptionWe consider the discrete symmetry’s dressing chains of the nonlinear Schrödinger equation (NLS) and Davey–Stewartson equations (DS). The modified NLS (mNLS) equation and the modified DS (mDS) equations are obtained. The explicitly reversible Bäcklund autotransformations for the mNLS and mDS equations are constructed. We demonstrate discrete symmetry’s conjugate chains of the KP and DS models. The twodimensional generalization of the equation is obtained.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Scattering of solitons of the Klein–Gordon equation coupled to a classical particle
View Description Hide DescriptionLongtime asymptotics are established for finite energy solutions of the scalar Klein–Gordon equation coupled to a relativistic classical particle: any “scattering” solution is asymptotically a sum of a soliton and of a dispersive free wave packet as These asymptotics mean the nonlinear scattering of free wave packets by the soliton.

 FLUIDS


Transport of energy in dissipative advection phenomena
View Description Hide DescriptionA study of the distribution of energy among the different scales is performed for several systems in fluid mechanics, including the Navier–Stokes, magnetohydrodynamics and active scalars equations. It is found that all these systems possess a common structure which enables us to deduce how the energy introduced by the forcing is transferred to the scales present in the flow. It is also shown that in special cases an energy cascade will occur. The limits of this method are also considered.

 STATISTICAL PHYSICS


The partial averaging method
View Description Hide DescriptionThe partial averaging technique is defined and used in conjunction with the random series implementation of the Feynman–Kaç formula. It enjoys certain properties such as good rates of convergence and convergence for potentials with coulombic singularities. In this work, I introduce the reader to the technique and I analyze the basic mathematical properties of the method. I show that the method is convergent for all Kato class potentials that have finite Gaussian transform.

Limits of a Ginzburg–Landau model with codimensionone defects
View Description Hide DescriptionWe derive a nondimensional Ginzburg–Landau energy functional that does not use temperaturedependent scaling quantities. Using the machinery of gammaconvergence, the asymptotic limits of the functional are computed in the presence of thin defects centered around codimension 1 manifolds. We classify the defects into three groups depending on the norm of the defect potential. We show that each group has its own distinguished asymptotic limit as the thickness of the defect converges to zero.

 METHODS OF MATHEMATICAL PHYSICS


Twisted superYangians and their representations
View Description Hide DescriptionStarting with the superYangian based on we define twisted superYangians Only and can be defined, and appear to be isomorphic one with each other. We study their finitedimensional irreducible highest weight representations.
