On the self‐adjointness of the Lorentz generator for (: ϕ4 :)1 + 1

An alternative proof to that provided by Jaffe and Cannon of the self‐adjointness of the local Lorentz generator for the (: ϕ4 :)1 + 1 quantum field theory is given. The proof avoids the use of second‐order estimates and a singular perturbation theory.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70022/2/JMAPAQ-15-6-867-1.pd

Quadratic Fermion Interaction Hamiltonian

The interaction Hamiltonian λ ∫ :(0)(x)ψ(0)(x):g(x)dsx,g(x) ∊ S(Rs)λ∫:ψ̄(0)(x)ψ(0)(x):g(x)dsx,g(x)∊S(Rs) is studied. An ultraviolet cutoff is introduced. We remove this cutoff, and take the limit g → 1 in S(Rs)S(Rs), by working with the Heisenberg fields. The limiting fields are well defined on the Fock space associated with the bare mass m0. In the limit we get a new representation of the canonical anticommutation relations which is given by a (generalized) Bogoliubov transformation. The new representation is not always unitarily equivalent to the bare mass Fock representations.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70542/2/JMAPAQ-12-7-1414-1.pd

Properties of the (: ϕ4 :)1 + 1 interaction Hamiltonian

Using a convergent expansion of the resolvent of the Hamiltonian H = H0+λV,V = ∫ dx×(x):φ4:(x),g(x) ∊ C0∞,g(x) ≥ 0H=H0+λV,V=∫dx×(x):φ4:(x),g(x)∊C0∞,g(x)⩾0, we give a simple proof of (a) the self‐adjointness of the Hamiltonian and (b) the volume independent lower bound of the vacuum energy per unit volume. Also, we obtain some coupling constant analyticity properties of the Hamiltonian, and the limit (H0+λν−z)−1→(H0−−z)−1(H0+λν−z)−1→(H0−−z)−1, z ∊ρ(H0) in norm as ∣λ∣→0 uniformly in {λ:∣argλ∣<π}{λ:∣argλ∣<π}.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70743/2/JMAPAQ-15-6-861-1.pd

Equilibrium of stellar dynamical systems in the context of the Vlasov-Poisson model

This short review is devoted to the problem of the equilibrium of stellar dynamical systems in the context of the Vlasov-Poisson model. In a first part we will review some classical problems posed by the application of the Vlasov-Poisson model to the astrophysical systems like globular clusters or galaxies. In a second part we will recall some recent numerical results which may give us some some quantitative hints about the equilibrium state associated to those systems.Comment: 8 pages, no figures, accepted in Communications in Nonlinear Science and Numerical Simulatio

Asymptotic and optimal Liouville properties for Wolff type integral systems

This article examines the properties of positive solutions to fully nonlinear systems of integral equations involving Hardy and Wolff potentials. The first part of the paper establishes an optimal existence result and a Liouville type theorem for the integral systems. Then, the second part examines the decay rates of positive bound states at infinity. In particular, a complete characterization of the asymptotic properties of bounded and decaying solutions is given by showing that such solutions vanish at infinity with two principle rates: the slow decay rates and the fast decay rates. In fact, the two rates can be fully distinguished by an integrability criterion. As an application, the results are shown to carry over to certain systems of quasilinear equations.Comment: 28 pages, author's final version incorporating reviewer comments and suggestion

Cooling down Levy flights

Let L(t) be a Levy flights process with a stability index \alpha\in(0,2), and U be an external multi-well potential. A jump-diffusion Z satisfying a stochastic differential equation dZ(t)=-U'(Z(t-))dt+\sigma(t)dL(t) describes an evolution of a Levy particle of an `instant temperature' \sigma(t) in an external force field. The temperature is supposed to decrease polynomially fast, i.e. \sigma(t)\approx t^{-\theta} for some \theta>0. We discover two different cooling regimes. If \theta<1/\alpha (slow cooling), the jump diffusion Z(t) has a non-trivial limiting distribution as t\to \infty, which is concentrated at the potential's local minima. If \theta>1/\alpha (fast cooling) the Levy particle gets trapped in one of the potential wells

Uniqueness results for an ODE related to a generalized Ginzburg-Landau model for liquid crystals

We study a singular nonlinear ordinary differential equation on intervals {[}0, R) with R <= +infinity, motivated by the Ginzburg-Landau models in superconductivity and Landau-de Gennes models in liquid crystals. We prove existence and uniqueness of positive solutions under general assumptions on the nonlinearity. Further uniqueness results for sign-changing solutions are obtained for a physically relevant class of nonlinearities. Moreover, we prove a number of fine qualitative properties of the solution that are important for the study of energetic stability

Statistical mechanics approach to the phase unwrapping problem

The use of Mean-Field theory to unwrap principal phase patterns has been recently proposed. In this paper we generalize the Mean-Field approach to process phase patterns with arbitrary degree of undersampling. The phase unwrapping problem is formulated as that of finding the ground state of a locally constrained, finite size, spin-L Ising model under a non-uniform magnetic field. The optimization problem is solved by the Mean-Field Annealing technique. Synthetic experiments show the effectiveness of the proposed algorithm

Renormalization of potentials and generalized centers

We generalize the Riesz potential of a compact domain in $\mathbb{R}^{m}$ by introducing a renormalization of the $r^{\alpha-m}$-potential for $\alpha\le0$. This can be considered as generalization of the dual mixed volumes of convex bodies as introduced by Lutwak. We then study the points where the extreme values of the (renormalized) potentials are attained. These points can be considered as a generalization of the center of mass. We also show that only balls give extreme values among bodied with the same volume.Comment: Adv. Appl. Math. 48 (2012), 365--392 Figure 11 has been corrected after publication. Theorem 3.12 and the exposition of Lemma 2.15 are modified in version