Commensurate-incommensurate phase transitions in one-dimensional chains

We consider one-dimensional systems of classical particles whose potential energy has the form: $$W_{\alpha ,\gamma } = \sum {[\alpha V(x_n )} + F(x_n - x_{n - 1} C\gamma )]$$ The limit of the Gibbs state as T→0 is described in terms of invariant measures of two-dimensional mappings which are constructed with the help ofW _{α, γ}. The dependence of these measures on parametersα, γ is investigated.     

A Strong Central Limit Theorem for a Class of Random Surfaces

This paper is concerned with d = 2 dimensional lattice field models with action ${V(\nabla\phi(\cdot))}$ , where ${V : \mathbf{R}^d \rightarrow \mathbf{R}}$ is a uniformly convex function. The fluctuations of the variable ${\phi(0) - \phi(x)}$ are studied for large |x| via the generating function given by ${g(x, \mu) = \ln \langle e^{\mu(\phi(0) - \phi(x))}\rangle_{A}}$ . In two dimensions ${g'' (x, \mu) = \partial^2g(x, \mu)/\partial\mu^2}$ is proportional to ${\ln\vert x\vert}$ . The main result of this paper is a bound on ${g''' (x, \mu) = \partial^3 g(x, \mu)/\partial \mu^3}$ which is uniform in ${\vert x \vert}$ for a class of convex V. The proof uses integration by parts following Helffer–Sjöstrand and Witten, and relies on estimates of singular integral operators on weighted Hilbert spaces.     

Superstable interactions in classical statistical mechanics

We consider classical systems of particles inv dimensions. For a very large class of pair potentials (superstable lower regular potentials) it is shown that the correlation functions have bounds of the form $$\varrho (x_1 ,...,x_n ) \leqq \xi ^n$$ . Using these and further inequalities one can extend various results obtained by Dobrushin and Minlos [3] for the case of potentials which are non-integrably divergent at the origin. In particular it is shown that the pressure is a continuous function of the density. Infinite system equilibrium states are also defined and studied by analogy with the work of Dobrushin [2a] and of Lanford and Ruelle [11] for lattice gases.     

Time decay of solutions to the cauchy problem for time-dependent Schrödinger-Hartree equations

We consider the time-dependent Schrödinger-Hartree equation (1) $$iu_t + \Delta u = \left( {\frac{1}{r}*|u|^2 } \right)u + \lambda \frac{u}{r},(t, x) \in \mathbb{R} \times \mathbb{R}^3 ,$$ (2) $$u(0,x) = \phi (x) \in \Sigma ^{2,2} ,x \in \mathbb{R}^3 ,$$ where λ≧0 and \(\Sigma ^{2,2} = \{ g \in L^2 ;\parallel g\parallel _{\Sigma ^{2,2} }^2 = \sum\limits_{|a| \leqq 2} {\parallel D^a g\parallel _2^2 + \sum\limits_{|\beta | \leqq 2} {\parallel x^\beta g\parallel _2^2< \infty } } \} \) . We show that there exists a unique global solutionu of (1) and (2) such that $$u \in C(\mathbb{R};H^{1,2} ) \cap L^\infty (\mathbb{R};H^{2,2} ) \cap L_{loc}^\infty (\mathbb{R};\Sigma ^{2,2} )$$ with $$u \in L^\infty (\mathbb{R};L^2 ).$$ Furthermore, we show thatu has the following estimates: $$\parallel u(t)\parallel _{2,2} \leqq C,a.c. t \in \mathbb{R},$$ and $$\parallel u(t)\parallel _\infty \leqq C(1 + |t|)^{ - 1/2} ,a.e. t \in \mathbb{R}.$$      

Percival variational principle for invariant measures and commensurate-incommensurate phase transitions in one-dimensional chains

We prove for a one-dimensional system of classical particles with potential energy, $$U_{\alpha ,\gamma } = \sum\limits_n {\left[ {\alpha V(x_n ) + F(x_{n + 1} - x_n - \gamma )} \right]} $$ , the existence of such a smooth function γ(α), 0≦α≦α₀(ω) that the system with potential energyU _αγ(α) has the equilibrium state at the temperatureT=0. This is the incommensurate phase with the ratio of periods equal to the prescribed irrational number ω, badly approximated by rational ones. A simple geometric condition for the invariant curve of the corresponding dynamical system is established under which it is the support of the invariant measure minimizing Percival's energy functional.     

Simulating spin-\frac{3}{2} particles at colliders

Support for interactions of spin- $\frac{3}{2}$ particles is implemented in the FeynRules and ALOHA packages and tested with the MadGraph 5 and CalcHEP event generators in the context of three phenomenological applications. In the first, we implement a spin- $\frac{3}{2}$ Majorana gravitino field, as in local supersymmetric models, and study gravitino and gluino pair-production. In the second, a spin- $\frac{3}{2}$ Dirac top-quark excitation, inspired from compositeness models, is implemented. We then investigate both top-quark excitation and top-quark pair-production. In the third, a general effective operator for a spin- $\frac{3}{2}$ Dirac quark excitation is implemented, followed by a calculation of the angular distribution of the s-channel production mechanism.     

On the existence of Feigenbaum's fixed point

We give a proof of the existence of aC ², even solution of Feigenbaum's functional equation $$g{\text{(}}x) = - \lambda _0^{ - 1} g{\text{(}}g( - \lambda _0 x)),g{\text{(0) = 1,}}$$ whereg is a map of [?1, 1] into itself. It extends to a real analytic function over ?.     

One-dimensional random Ising systems with interaction decayr −(1+ɛ): A convergent cluster expansion

WE consider a one-dimensional random Ising model with Hamiltonian $$H = \sum\limits_{i\ddag j} {\frac{{J_{ij} }}{{\left| {i - j} \right|^{1 + \varepsilon } }}S_i S_j } + h\sum\limits_i {S_i } $$ , where ε>0 andJ _ij are independent, identically distributed random variables with distributiondF(x) such that i) $$\int {xdF\left( x \right) = 0} $$ , ii) $$\int {e^{tx} dF\left( x \right)< \infty \forall t \in \mathbb{R}} $$ . We construct a cluster expansion for the free energy and the Gibbs expectations of local observables. This expansion is convergent almost surely at every temperature. In this way we obtain that the free energy and the Gibbs expectations of local observables areC ^∞ functions of the temperature and of the magnetic fieldh. Moreover we can estimate the decay of truncated correlation functions. In particular for every ε′>0 there exists a random variablec(ω)m, finite almost everywhere, such that $$\left| {\left\langle {s_0 s_j } \right\rangle _H - \left\langle {s_0 } \right\rangle _H \left\langle {s_j } \right\rangle _H } \right| \leqq \frac{{c\left( \omega \right)}}{{\left| j \right|^{1 + \varepsilon - \varepsilon '} }}$$ , where 〈 〉 _H denotes the Gibbs average with respect to the HamiltonianH.     

Asymptotic observables on scattering states

In quantum mechanical potential scattering theory we use selected observables to describe the asymptotic properties of scattering states for long times. E.g., we show for the position and momentum operators that for Ωε?^cont(H), $$\left( {m\frac{x}{t} - p} \right)e^{ - iHt} \Psi \to 0,$$ and that the set of outgoing states is absorbing. This is obtained easily without any detailed analysis of the interacting time evolution. The class of forces includes highly singular and very long range potentials. The results may serve as an intermediate step in a proof of asymptotic completeness; as a particular application we present a simple proof of completeness for Coulomb systems.     

Non-existence of spontaneous magnetization in a one-dimensional Ising ferromagnet

It is proved that an infinite linear chain of spins μ _j =±1, with an interaction energy $$H = - \Sigma J(i - j)\mu _i \mu _j $$ has zero spontaneous magnetization at all finite temperatures, provided thatJ (n) is non-negative and that $$H = - \Sigma J(i - j)\mu _i \mu _j $$ . This shows that a theorem ofRuelle, establishing the absence of long-range order when the sum Σn J (n) converges, is not the best possible.     

The mass gap for theP(φ)2 quantum field model with a strong external field

We consider theP(φ)₂ hamiltonian whose interaction density is given by $$\lambda P(\phi (x)) + \mu \phi (x)^k $$ wherek is odd and 1≦kP. For sufficiently large μ we show that there is a gap in the energy spectrum. In addition we obtain new regions of analyticity in λ and μ for the Schwinger functions and the pressure.     

One-Dimensional Disordered Quantum Mechanics and Sinai Diffusion with Random Absorbers

We study the one-dimensional Schrödinger equation with a disordered potential of the form $$\begin{aligned} V (x) = \phi (x)^2+\phi '(x) + \kappa (x) \end{aligned}$$ where $\phi (x)$ is a Gaussian white noise with mean $\mu g$ and variance $g$ , and $\kappa (x)$ is a random superposition of delta functions distributed uniformly on the real line with mean density $\rho $ and mean strength $v$ . Our study is motivated by the close connection between this problem and classical diffusion in a random environment (the Sinai problem) in the presence of random absorbers: $\phi (x)$ models the force field acting on the diffusing particle and $\kappa (x)$ models the absorption properties of the medium in which the diffusion takes place. The focus is on the calculation of the complex Lyapunov exponent $ \varOmega (E) = \gamma (E) - \mathrm{i}\pi N(E) $ , where $N$ is the integrated density of states per unit length and $\gamma $ the reciprocal of the localisation length. By using the continuous version of the Dyson–Schmidt method, we find an exact formula, in terms of a Hankel function, in the particular case where the strength of the delta functions is exponentially-distributed with mean $v=2g$ . Building on this result, we then solve the general case— in the low-energy limit— in terms of an infinite sum of Hankel functions. Our main result, valid without restrictions on the parameters of the model, is that the integrated density of states exhibits the power law behaviour $$\begin{aligned} N(E) \underset{E\rightarrow 0+}{\sim } E^\nu \quad \hbox {where } \quad \nu =\sqrt{\mu ^2+2\rho /g}. \end{aligned}$$ This confirms and extends several results obtained previously by approximate methods.     

Blow Up Dynamics for Equivariant Critical Schrödinger Maps

For the Schrödinger map equation \({u_t = u \times \triangle u \, {\rm in} \, \mathbb{R}^{2+1}}\) , with values in S ², we prove for any \({\nu > 1}\) the existence of equivariant finite time blow up solutions of the form \({u(x, t) = \phi(\lambda(t) x) + \zeta(x, t)}\) , where \({\phi}\) is a lowest energy steady state, \({\lambda(t) = t^{-1/2-\nu}}\) and \({\zeta(t)}\) is arbitrary small in \({\dot H^1 \cap \dot H^2}\) .     

Exterior complex scaling and the AC-Stark effect in a Coulomb field

By means of the exterior complex scaling of B. Simon an existence proof of resonances is given for the time-dependent Schrödinger equation $$i\frac{{\partial \psi }}{{\partial t}} = - \left( { - \Delta + V + \mu x_1 \cos \omega t} \right)\psi ,$$ whereV belongs to a class of potentials which includes the Coulomb one. The resonance width is given by the Fermi Golden Rule to second order perturbation theory and is nonzero for μ small and almost every ε.     

Complete mass spectra ofq\bar q andQ\bar q mesons with improved Bethe-Salpeter dynamics of confinementmesons with improved Bethe-Salpeter dynamics of confinement

The Bethe-Salpeter (bs) dynamics of harmonic confinement developed byanm and collaborators over the last three years and already applied with considerable experimental success to various hadron spectra and coupling structures has been significantly improved through (i) a more exact treatment of a certain momentum-dependent operator \(\hat Q_q \) appearing in thebs equation, using the techniques of SO (2, 1) Lie algebra, and (ii) a sharpened definition of theqcd Coulomb term, so as to yield unambiguous values for different flavour sectors. The resulting mass spectra of light \((q\bar q)\) meson towers and semi-heavy \((Q\bar q)\) quarkonia which are most sensitive to the improved treatment of \(\hat Q_q \) , reveal excellent agreement with experiment, one in which only slight changes in the reduced spring constant \((\tilde \omega )\) and quark masses (m _q ) over the earlier parametrizations are involved. These changes are however found to have a negligible effect on the (already good) numerical values of the other predictions (electroweak and pionic couplings) depending on the \(q\bar q\) andqqq wave functions. A critical assessment of the strong and weak points of this method is madevis-a-vis other related approaches.     

Truncated Connectivities in a Highly Supercritical Anisotropic Percolation Model

We consider an anisotropic bond percolation model on $\mathbb{Z}^{2}$ , with p=(p _h ,p _v )∈[0,1]², p _v >p _h , and declare each horizontal (respectively vertical) edge of $\mathbb{Z}^{2}$ to be open with probability p _h (respectively p _v ), and otherwise closed, independently of all other edges. Let $x=(x_{1},x_{2}) \in\mathbb{Z}^{2}$ with 0<x ₁<x ₂, and $x'=(x_{2},x_{1})\in\mathbb{Z}^{2}$ . It is natural to ask how the two point connectivity function $\mathbb{P}_{\mathbf{p}}(\{0\leftrightarrow x\})$ behaves, and whether anisotropy in percolation probabilities implies the strict inequality $\mathbb{P}_{\mathbf{p}}(\{0\leftrightarrow x\})>\mathbb{P}_{\mathbf {p}}(\{0\leftrightarrow x'\})$ . In this note we give an affirmative answer in the highly supercritical regime.     

Towards Asymptotic Completeness of Two-Particle Scattering in Local Relativistic QFT

We consider the problem of existence of asymptotic observables in local relativistic theories of massive particles. Let ${\tilde{p}_1}$ and ${\tilde{p}_2}$ be two energy-momentum vectors of a massive particle and let ${\Delta}$ be a small neighbourhood of ${\tilde{p}_1 + \tilde{p}_2}$ . We construct asymptotic observables (two-particle Araki–Haag detectors), sensitive to neutral particles of energy-momenta in small neighbourhoods of ${\tilde{p}_1}$ and ${\tilde{p}_2}$ . We show that these asymptotic observables exist, as strong limits of their approximating sequences, on all physical states from the spectral subspace of ${\Delta}$ . Moreover, the linear span of the ranges of all such asymptotic observables coincides with the subspace of two-particle Haag–Ruelle scattering states with total energy-momenta in ${\Delta}$ . The result holds under very general conditions which are satisfied, for example, in ${\lambda{\phi}_{2}^{4}}$ . The proof of convergence relies on a variant of the phase-space propagation estimate of Graf.     

Lattice systems with a continuous symmetry

We consider perturbations of a massless Gaussian lattice field on ? ^d ,d≧3, which preserves the continuous symmetry of the Hamiltonian, e.g., $$ - H = \sum\limits_{< x,y > } {(\phi _x - \phi _y )^2 + T(\phi _x - \phi _y )^4 ,\phi _x \in \mathbb{R}.} $$ It is known that for allT>0 the correlation functions in this model do not decay exponentially. We derive a power law upper bound for all (truncated) correlation functions. Our method is based on a combination of the log concavity inequalities of Brascamp and Lieb, reflection positivity and the Fortuin, Kasteleyn and Ginibre (F.K.G.) inequalities.     

Correlation functions and the uniqueness of the state in classical statistical mechanics

A general criterion is derived which assures the uniqueness of the state of a classical statistical mechanical system in terms of a given system of correlation functions. The criterion is \(\sum\limits_k {(m_{k + j}^A )} ^{ - 1/k} = \infty\) for allj and all bounded setsA, where $$m_k^A = (k!)^{ - 1} \int\limits_A \cdots \int\limits_A {\varrho _k } (x_1 , \ldots ,x_k )dx_1 \ldots dx_1 .$$      

Asymptotic behavior of solutions to certain nonlinear Schrödinger-Hartree equations

The asymptotic behavior of solutions to the Cauchy problem for the equation $$i\psi _\imath = \frac{1}{2}\Delta \psi - \upsilon (\psi )\psi , \upsilon = r^{ - 1} *|\psi |^2 ,$$ and for systems of similar form, is studied. It is shown that the norms $$\parallel \psi (t)\parallel _{L_2 (|x| \leqq R)}^2 + \parallel \nabla \psi (t)\parallel _{L_2 (|x| \leqq R)}^2 $$ are integrable in time for any fixedR>0, from which it follows that $$\mathop {\lim }\limits_{t \to \infty } \parallel \psi (t)\parallel _{L_2 (|x| \leqq R)} = 0.$$ \] Nevertheless, it is established that anL ₂-scattering theory is impossible.