Bethe's Equation Is Incomplete for the XXZ Model at Roots of Unity

We demonstrate for the six vertex and XXZ model parameterized by = –(q+q^-1)/2±1 that when q^2N=1 for integer N2 the Bethe's ansatz equations determine only the eigenvectors which are the highest weights of the infinite dimensional sl₂ loop algebra symmetry group of the model. Therefore in this case the Bethe's ansatz equations are incomplete and further conditions need to be imposed in order to completely specify the wave function. We discuss how the evaluation parameters of the finite dimensional representations of the sl₂ loop algebra can be used to complete this specification.     

Construction of modular branching functions from Bethe's equations in the 3-state Potts chain

We use the single-particle excitation energies and the completeness rules of the 3-state antiferromagnetic Potts chain, which have been obtained from Bethe's equation, to compute the modular invariant partition function. This provides a fermionic construction for the branching functions of theD ₄ representation ofZ ₄ parafermions which complements the bosonic constructions. It is found that there are oscillations in some of the correlations and a new connection with the field theory of the Lee-Yang edge is presented.     

Fermion Fields,Boson Fields,and Gravitational Field

If a tetrad theory is derivable from a variational principle with a Lagrangian ?? of the form ?? = ??_F+??_M 6 tetrad components will be defined by the vacuum equations if the energy momentum tensor is symmetric. Therefore, we look for a realisation of a programme proposed in a little different way by TREDER according to which the 16 tetrad field equations should degenerate to 10 equations for the Riemannian metric if boson fields are the only source of the gravitational field.     

Bethe-Ansatz solution of a model for a mixed valent impurity with two magnetic configurations: Groundstate and thermodynamics

We consider a mixed valence impurity with two magnetic configurations of spinJ ₂ andJ ₁=J ₂±1/2, respectively, coupled bys-wave conduction electrons via a hybridization matrix element. The model contains theU limit of the non-degenerate Anderson model and the Kondo exchange Hamiltonian for arbitrary spin as special cases. The model is solved by Bethe's ansatz and the groundstate and the thermodynamic properties are discussed. The Kondo limit and the highT perturbation expansion are extracted from the thermodynamic Bethe-ansatz equations. The ground state is magnetic if neitherJ ₁ norJ ₂ is a singlet.Heisenberg fellow of the Deutsche Forschungsgemeinschaft     

Bethe-Peierls theory for the dilute RKKY spin glass

The mean field theory of the spin-glass transition due to Edwards and Anderson is not directly applicable to dilute-alloy spin glasses because of the prevalence of strong exchange interactions, such that |J|k _BT_g. The effects of strong interactions are better described by the Bethe-Peierls approximation. Applied to classical dilute magnets with RKKY interaction, Bethe's method gives results consistent with the concentration scaling law of Blandin and Souletie, in particular that the glass temperatureT _g be linear in concentration. However, the critical exponents are just those of the mean field theory. The calculated variation of susceptibility with temperature just belowT _g is not in agreement with experiment, and possible reasons for this are discussed.     

Limiting Behavior of Eigenvectors of Large Wigner Matrices

A new form of empirical spectral distribution of a Wigner matrix W _n with weights specified by the eigenvectors is defined and it is then shown to converge with probability one to the semicircular law. Moreover, central limit theorem for linear spectral statistics defined by the eigenvectors and eigenvalues is also established under some moment conditions, which suggests that the eigenvector matrix of W _n is close to being Haar distributed.     

Completeness of a system of eigenvectors of quadratic operator sheaf in waveguide theory

The completeness of the system of eigenvectors H _n of a quadratic operator sheaf that occurs in the theory of electromagnetic waveguides is proven. The completeness of another system of eigenvectors of the spectral problem of waveguide theory that was considered in another formulation that was established earlier is used in the proof.     

Passive Advection and the Degenerate Elliptic Operators <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

 We prove estimates for the stationary state n-point functions at zero molecular diffusivity in the Kraichnan model [13]. This is done by proving upper bounds for the heat kernels and Green's functions of the degenerate elliptic operators M _n that occur in the Hopf equations for the n-point functions. Received: 25 August 2001 / Accepted: 30 September 2002 Published online: 20 January 2003 Communicated by A. Kupiainen     

Comprehensive study of the correlations that exist among the members of the [n]cyclacene series and the Möbius[n]cyclacene series

Cyclacenes are the smallest substructures of carbon nanotubes used in modelling studies. The systematics that exists between Hückel molecular orbital eigenvalues and eigenvectors of cyclacenes are delineated. This study of cyclacenes combines the interconnection of concepts of complementarity theorem, characteristic and matching polynomial recursion equations, embedding, greater than twofold symmetry and doubly degenerate eigenvalues, open-shell singlet character, and pairing theorem. Proof that cyclacenes have more open-shell (diradical) character than do polyacenes is also provided by the sum total of this work. Mirror-plane scission of even-ring cyclacenes gives linear polyacene fragments. This shows that the properties of the successor linear polyacene must be contained in the precursor cyclacene. Corresponding Möbius[n]cyclacene isomers display contrasting and unusual comparative properties. A partial list of contrasting properties include alternant (cyclacenes) versus nonalternant (Möbius[n]cyclacene) polyenes, presence of Hamiltonian circuits in Möbius[n]cyclacene and presence of oscillatory electronic properties in cyclacenes.     

On the phase transitions for the electronic excitations in semiconductors

A model Hamiltonian for a system of interacting electrons, holes and Wannier excitons is derived. This system of electronic excitations is assumed to be in a quasi-equilibrium state. With the aid of Bogolubov's variational principal the thermodynamic potential is calculated. Using the most general mean-field Hamiltonian as a trial Hamiltonian, a set of coupled integral equations is obtained for the self-energies. These equations are solved numerically for equal effective masses of the electrons and holes. Below a critical temperature ofk _B T _c0.65E _ex ^b whereE _ex ^b is the exciton binding energy, we find a first order phase transition from an exciton rich phase into a degenerate electron-hole phase. The mechanical and thermal stability of both phases is proven. Below a critical temperaturek _B T _c0.11E _ex ^b the exciton system becomes degenerate (Bose-Einstein condensation). A complete phase diagram of these three phases is given.This is a project of the Sonderforschungsbereich Frankfurt/Darmstadt, financed by special funds of the Deutsche Forschungsgemeinschaft     

Rational Surfaces Associated with Affine Root Systems¶and Geometry of the Painlevé Equations

We present a geometric approach to the theory of Painlevé equations based on rational surfaces. Our starting point is a compact smooth rational surface X which has a unique anti-canonical divisor D of canonical type. We classify all such surfaces X. To each X, there corresponds a root subsystem of E ⁽¹⁾ ₈ inside the Picard lattice of X. We realize the action of the corresponding affine Weyl group as the Cremona action on a family of these surfaces. We show that the translation part of the affine Weyl group gives rise to discrete Painlevé equations, and that the above action constitutes their group of symmetries by B?cklund transformations. The six Painlevé differential equations appear as degenerate cases of this construction. In the latter context, X is Okamoto's space of initial conditions and D is the pole divisor of the symplectic form defining the Hamiltonian structure. Received: 18 September 1999 / Accepted: 29 January 2001     

A Priori Estimates for the Free-Boundary 3D Compressible Euler Equations in Physical Vacuum

We prove a priori estimates for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) = C _γ ρ ^γ for γ > 1. The vacuum condition necessitates the vanishing of the pressure, and hence density, on the dynamic boundary, which creates a degenerate and characteristic hyperbolic free-boundary system to which standard methods of symmetrizable hyperbolic equations cannot be applied.     

Periodic and solitary wave solutions of the three-wave problem. A different approach

We consider the inverse problem for a three-wave interaction system in a manner different from Zakharovet al. and of Kaup. Our method is an adaptation of the technique due to Date to a 3 × 3 Lax pair. The analysis leads to a system of ordinary nonlinear equations for the _ivariables linearizable through a suitable definition of differential on a Riemann surface. Next, in the degenerate case, when the _iare equal in pairs, we prove that such a set of equations is exactly integrable and leads to solitary solutions.     

Flow-equations for Hamiltonians

Flow-equations are introduced in order to bring Hamiltonians closer to diagonalization. It is characteristic for these equations that matrix-elements between degenerate or almost degenerate states do not decay or decay very slowly. In order to understand different types of physical systems in this framework it is probably necessary to classify various types of these degeneracies and to investigate the corresponding physical behavior. In general these equations generate many-particle interactions. However, for an n-orbital model the equations for the two-particle interaction are closed in the limit of large n. Solutions of these equations for a one-dimensional model are considered. There appear convergency problems, which are removed, if instead of diagonalization only a block-diagonalization into blocks with the same number of quasiparticles is performed.     

Bianchi Type V magnetized string dust cosmological models with Petrov-type degenerate

Bianchi Type V massive string cosmological models with free gravitational field of Petrov Type degenerate in the presence of magnetic field with variable magnetic permeability are investigated. The magnetic field is due to an electric current produced along the x-axis. The F ₂₃ is the only non-vanishing component of electromagnetic field tensor F _ij . Maxwell’s equations F _[ij;k] = 0 and F _;j ^ij = 0 are satisfied by F ₂₃ = constant. The behaviour of the model in the presence and absence of magnetic field and other physical aspects are also discussed.      

Shear-free,twisting Einstein-Maxwell metrics in the Newman-Penrose formalism

The problem of finding algebraically special solutions of the vacuum Einstein-Maxwell equations is investigated using the spin coefficient formalism of Newman and Penrose. The general case, in which the degenerate null vectors are not hypersurface orthogonal, is reduced to a problem of solving five coupled differential equations that are no longer dependent on the affine parameter along the degenerate null directions. It is shown that the most general regular, shearfree, nonradiating solution of these equations is the Kerr-Newman metric.Based in part on a doctoral thesis submitted to the University of Pittsburgh (1970) while the author was NASA Predoctoral Trainee. Research also supported in part by the National Science Foundation under Grant GP-19378.     

Molecular viewpoint on high-temperature superconductivity Importance of orbital degeneracy

It is shown that the antisymmetrized geminal power wavefunction (AGP) in the macroscopic limit and the Bardeen-Cooper-Schrieffer (BCS) supercon-ductivity model with fixed mean number of electrons coincide to arbitrary order in deviations from the extreme-type function which is considered as the carrier of the superconductivity property. Variational equations for the AGP in the macroscopic limit are formulated in terms of two sets of parameters, ∈ _i and Δ_i , which under simplifying assumptions reduce to eigenvalues of the open-shell Roothaan one-electron Hamiltonian and to the BCS energy gap parameter, respectively. The superconducting state is shown to be stable for the solution of these equations with a macroscopic number of non-zero Δ_i and of degenerate ∈ _i =∈_F at the Fermi level ∈_F. The macroscopic contribution to the maximal pair occupation number which is responsible for the superconductivity is expressed as a mean value of Δ_i ²/[(∈ _i ?∈_F)²+Δ_i ²]. The formulated non-zero temperature version of the equations for ∈ _i , Δ_i is able to describe the superconducting phase transition. On this ground the necessary condition of stabilization of the superconducting state is formulated that is the existence of the macroscopic-fold near-degenerate and almost half-filled level. As is shown it is realized in the energy band structure of doped fullerides, copper oxide ceramics and perovskite-type crystals, e.g. BaBiO₃. The additional requirement of negativity of exchange interelectron-interaction integrals may be satisfied not only by the known vibronic mechanism but also, as is demonstrated, by the polarization potential of an environment in a plane layer of stratum structures.     

Local dynamics and gravitational collapse of a self-gravitating magnetized Fermi gas

We use the Bianchi-I spacetime to study the local dynamics of a magnetized self-gravitating Fermi gas. The set of Einstein–Maxwell field equations for this gas becomes a dynamical system in a 4D phase space. We consider a qualitative study and examine numeric solutions for the degenerate zero temperature case. All dynamic quantities exhibit similar qualitative behavior in the 3D sections of the phase space, with all trajectories reaching a stable attractor whenever the initial expansion scalar H ₀ is negative. If H ₀ is positive the trajectories end up in a curvature singularity that can be, depending on initial conditions, isotropic or anisotropic. In particular, if the initial magnetic field intensity is sufficiently large the collapsing singularity will always be anisotropic and pointing in the same direction of the field.     

A new hierarchy of coupled degenerate hamiltonian equations

A new hierarchy of pairs of coupled nonlinear evolution equations is proposed. The first pair of coupled equations can be reduced to the single equation w_tt - w_tx ± 4e^w = 0. It is shown that the equations in this hierarchy possess an infinite number of conserved densities, and that a degenerate symplectic structure can be introduced to the whole hierarchy.     

Ideals generated by traces or by supertraces in the symplectic reflection algebra H1,ν (I2(2m + 1))

For each complex number ν, an associative symplectic reflection algebra ? :=?H_1,ν (I₂(2m?+?1)), based on the group generated by root system I₂(2m?+?1), has an m-dimensional space of traces and an (m?+?1)-dimensional space of supertraces. A (super)trace sp is said to be degenerate if the corresponding bilinear (super)symmetric form B_sp(x, y)?=?sp(xy) is degenerate. We find all values of the parameter ν for which either the space of traces contains a degenerate nonzero trace or the space of supertraces contains a degenerate nonzero supertrace and, as a consequence, the algebra ? has a two-sided ideal of null-vectors. The analogous results for the case H_1,ν1,ν2(I₂(2m)) are also presented.