On the Bardeen-Cooper-Schrieffer integral equation in the theory of superconductivity

The Bardeen-Cooper-Schrieffer integral equation with a positive kernel is studied in full generality. It is shown that, there exists a unique finite transition temperature, T _c so that, if T _c ,the equation possesses a positive solution, representing the onset of the superconducting phase, while if T>T _c ,the only solution of the equation is the trivial one, indicating the occurrence of the normal phase. Moreover, it is demonstrated that such a positive solution may be approximated by a sequence of solutions of the equation restricted on bounded domains. This latter result provides a useful computational scheme for the problem.     

The dynamical entropy of the quantum Arnold cat map

We present a rigorous computation of the dynamical entropyh of the quantum Arnold cat map. This map, which describes a flow on the noncommutative two-dimensional torus, is a simple example of a quantum dynamical system with optimal mixing properties, characterized by Lyapunov exponents ± 1n ⁺, ⁺ > 1. We show that, for all values of the quantum deformation parameter,h coincides with the positive Lyapunov exponent of the dynamics.     

Defining quantum dynamical entropy

We propose an elementary definition of the dynamical entropy for a discrete-time quantum dynamical system. We apply our construction to classical dynamical systems and to the shift on a quantum spin chain. In the first case, we recover the Kolmogorov-Sinai invariant and, for the second, we find the mean entropy of the invariant state plus the logarithm of the dimension of the single-spin space.     

Phase-space moment-equation model of highly relativistic electron-beams in plasma-wakefield accelerators

We formulate a new procedure for modelling the transverse dynamics of relativistic electron beams with significant energy spread when injected into plasma-based accelerators operated in the blow-out regime. Quantities of physical interest, such as the emittance, are furnished directly from solution of phase space moment equations formed from the relativistic Vlasov equation. The moment equations are closed by an Ansatz, and solved analytically for prescribed wakefields. The accuracy of the analytic formulas is established by benchmarking against the results of a semi-analytic/numerical procedure which is described within the scope of this work, and results from a simulation with the 3D quasi-static PIC code HiPACE.     

Deformation quantization of nonholonomic almost Kähler models and Einstein gravity

Nonholonomic distributions and adapted frame structures on (pseudo) Riemannian manifolds of even dimension are employed to build structures equivalent to almost Kähler geometry and which allows to perform a Fedosov-like quantization of gravity. The nonlinear connection formalism that was formally elaborated for Lagrange and Finsler geometry is implemented in classical and quantum Einstein gravity.     

Time-asymptotic boson states from infinite mean field quantum systems coupled to the boson gas

By means of cocycle techniques in a recent paper, the global dynamics of mean field-boson couplings has been studied. Here, by restricting to the bosonic system the infinite time limit (t ) for very general initial states, one obtains time-asymptotic states on the bosonicC ^*-Weyl algebra, in which one partially rediscovers the collective ordering of the infinite mean field lattice.     

On the dynamics and the temperature states of the spin-boson model

Recently, the finite temperature states of the spin-boson model were studied. Here with different techniques in the case of level-splitting zero, the explicit expressions for both the dynamics and the equilibrium states are derived, namely Trotter's product formula and analyticity methods from perturbation theory are used.     

On the extremality of the disordered state for the Ising model on the Bethe lattice

We give a simple proof that the limit Ising Gibbs measure with free boundary conditions on the Bethe lattice with the forward branching ratio k2 is extremal if and only if is less or equal to the spin glass transition value, given by tanh( _c ^SG = 1/k.The work was partially supported by the NSF grant DMS 9504513.     

Stability and related properties of vacua and ground states

We consider the formal non-relativistic limit (nrl) of the :?⁴:_s+1 relativistic quantum field theory (rqft), where s is the space dimension. Following the work of R. Jackiw [R. Jackiw, in: A. Ali, P. Hoodbhoy (Eds.), Bég Memorial Volume, World Scientific, Singapore, 1991], we show that, for s = 2 and a given value of the ultraviolet cutoff κ, there are two ways to perform the nrl: (i) fixing the renormalized mass m² equal to the bare mass ; (ii) keeping the renormalized mass fixed and different from the bare mass . In the (infinite-volume) two-particle sector the scattering amplitude tends to zero as κ → ∞ in case (i) and, in case (ii), there is a bound state, indicating that the interaction potential is attractive. As a consequence, stability of matter fails for our boson system. We discuss why both alternatives do not reproduce the low-energy behaviour of the full rqft. The singular nature of the nrl is also nicely illustrated for s = 1 by a rigorous stability/instability result of a different nature.     

Long-range order in a simple model of interacting fermions

We consider a model of spinless fermions on a lattice, interacting through a nearest neighbor repulsion. In the half-filled band case and for dimensionsd 2, we rigorously prove that there is long-range order in some domain of the parameters=(k _B T)^–1 andt/U, wheret is the hopping amplitude of the particles,U the strength of their repulsion, and the inverse temperature. Our technique is based on the usual Peierls argument of classical statistical mechanics but fails for the groundstate. We discuss the specific difficulties introduced by the Fermi statistics.Work supported in part by U.S. NSF grant PHY 90-19433-A02.     

The free energy of systems with net charge

For quite general domains, the free energy density of a system with net charge differs in the thermodynamic limit from that of a neutral system by an electrostatic correction.     

An invariant formula for a star product with separation of variables

We give an invariant formula for a star product with separation of variables on a pseudo-Kähler manifold.     

Entropy,the central limit theorem and the algebra of the canonical commutation relation

The central limit theorem of Cushen and Hudson is reformulated on the algebra of the CCR. Namely, for a gauge invariant state , the weighted convolutions _n of the central limit tend to the quasi-free reduction _Q of pointwise. It is proved that if the initial relative entropy S(, _Q ) is finite, then S( _n , _Q ) goes to 0 and so _n – _Q 0. No restriction on the dimension of the test function space is made.     

Renormalization group method based on the ionization energy theory

Proofs are developed to explicitly show that the ionization energy theory is a renormalized theory, which mathematically exactly satisfies the renormalization group formalisms developed by Gell–Mann–Low, Shankar and Zinn-Justin. However, the cutoff parameter for the ionization energy theory relies on the energy-level spacing, instead of lattice point spacing in k-space. Subsequently, we apply the earlier proofs to prove that the mathematical structure of the ionization-energy dressed electron–electron screened Coulomb potential is exactly the same as the ionization-energy dressed electron–phonon interaction potential. The latter proof is proven by means of the second-order time-independent perturbation theory with the heavier effective mass condition, as required by the electron–electron screened Coulomb potential. The outcome of this proof is that we can derive the heat capacity and the Debye frequency as a function of ionization energy, which can be applied in strongly correlated matter and nanostructures.     

Duality for Jacobi Group Orbit Spaces and Elliptic Solutions of the WDVV Equations

From any given Frobenius manifold one may construct a so-called ’dual’ structure which, while not satisfying the full axioms of a Frobenius manifold, shares many of its essential features, such as the existence of a prepotential satisfying the Witten– Dijkgraaf–Verlinde–Verlinde equations of associativity. Jacobi group orbit spaces naturally carry the structure of a Frobenius manifold and hence there exists a dual prepotential. In this paper this dual prepotential is constructed and expressed in terms of the elliptic polylogarithm function of Beilinson and Levin.     

Finite energy sectors of the XY model

Continuing the earlier work on soliton sectors, we determine all finite energy representations of the XY model for almost all parameter values. In the interior of unique vacuum regions of parameters (i.e. the large external magnetic field region ||>1), the unique irreducible vacuum representation is the only finite energy representation.At the critical values of the parameters (||=1 as well as theXY symmetric case =0, ||1), there is an infinite number of mutually nonequivalent irreducible finite energy representations. Apart from the unique irreducible ground state representation and another associated irreducible representation, these infinite number of representations arise from an infinite number of nearly zero energy excitations of the ground state with a finite total energy and may be called infrared representations.In the remaining cases, as have been studied earlier, there are two additional irreducible finite energy representations besides two irreducible ground state representations and they are topological soliton sectors with different ground state limits in positive and negative spatial infinity. (For two exceptional values of parameters (, )=(0, ±1), they also become ground state representations.)     

On the meissner and abelian higgs effects

We provide an axiomatic Euclidean field-theoretic treatment of the Meissner and Abelian Higgs effects, which extends our previous method to a fully quantum theoretic model of a gauge field, with U(1) symmetry, coupled to a scalar field. Our main results are that the former effect implies the latter one, and ensues from the condition of off-diagonal long-range order.     

Abundance of translation invariant pure states on quantum spin chains

We construct a set of translation invariant pure states of a quantum spin chain, which is w -dense in the set of all translation invariant states of the chain. Each of the approximating states has exponential decay of correlations, and is the unique ground state of a finite range Hamiltonian with a spectral gap above the ground state energy.