Borel summability of the perturbation series in a hierarchical λ(▽φ)4 model

We prove Borel summability of the perturbation series for the dielectric constant and the free energy density for the hierarchical ()⁴ lattice model. Our methods are based on nonperturbative renormalization group analysis of the model.On leave from the Department of Mathematical Methods of Physics, Warsaw University, Poland.Supported in part by the Center for Interdisciplinary Research, Bielefeld University, Germany.     

The central limit theorem and the problem of equivalence of ensembles

In this paper we show that the local limit theorem is a consequence of the integral central limit theorem in the case of a Gibbs random field _t ,tZ corresponding to a finite range potential.We apply this theorem to show that the equivalence between Gibbs and canonical ensemble is a consequence of the integral central limit theorem and of very weak conditions on decrease of correlations.Research supported by a C.N.R.-Ak. Nauk. U.R.S.S. fellowship     

Time evolution of infinite classical systems with singular,long range,two body interactions

Existence of dynamics for infinitely many hard-spheres inv dimensions is proven in a set of full equilibrium measure.Singular unbounded perturbations are considered with pair potentials diverging as (x – a)^–, >2 anda is the hard-core diameter. Long range forces are allowed with potentials decreasing at infinity asx , <v. The result corrects and generalizes a proof given in a previous paper by the same authors.Research partially supported by a CNR fellowship Posit. 204530.Research partially supported by a CNR fellowship.     

Cauchy—Riemann conditions and point singularities of solutions to linearized shallow-water equations

Singular solutions with algebraic “square-root” type singularity of two-dimensional equations of shallow-water theory are propagated along the trajectories of the external velocity field on which the field satisfies the Cauchy-Riemann conditions. In other words, the differential of the phase flow is proportional to an orthogonal operator on such a trajectory. It turns out that, in the linear approximation, this fact is closely related to the effect of “blurring” of solutions of hydrodynamical equations; namely, a singular solution of the Cauchy problem for the linearized shallow-water equations preserves its shape exactly (i.e., is not blurred) if and only if the Cauchy-Riemann conditions are satisfied on the trajectory (of the external field) along which the perturbation is propagated.     

Time evolution of a quantum lattice system

It is shown that for a quantum lattice system associated with a Hamiltonian with a kinetic part and a potential sufficiently decreasing in the particle number, the time evolution can be described, under certain assumptions, by automorphisms of a suitable algebra.     

Two-dimensional wave equation with degeneration on the curvilinear boundary of the domain and asymptotic solutions with localized initial data

In a two-dimensional domain Ω ? R ², we consider the wave equation with variable velocity c(x ₁, x ₂) degenerating on the boundary Γ = ?Ω as the square root of the distance to the boundary, and construct an asymptotic solution of the Cauchy problem with localized initial data. This problem is related to the so-called “run-up problem” in tsunami wave theory. One main idea (also used by the authors in earlier papers in the one-dimensional case and the two-dimensional case with c ²(x ₁, x ₂) = x ₁) is that the (singular) curve Γ is a caustic of special type. We use this idea to introduce a generalization of the Maslov canonical operator covering the problem with degeneration and obtain efficient formulas for the asymptotic solutions.     

Asymptotics of localized solutions of the one-dimensional wave equation with variable velocity. II. Taking into account a source on the right-hand side and a weak dispersion

In the present (second) part of the paper, we study the asymptotic behavior of the solution of the Cauchy problem for a nonhomogeneous wave equation and also consider (instead of the wave equation) an equation with added fourth derivatives containing a small parameter, i.e., include the effects of weak dispersion. The research was supported by RFBR grant no. 08-01-00726 and by the scientific agreement between the Department of Physics of the University of Rome “La Sapienza“ and the A. Ishlinski Institute for Problems in Mechanics of Russian Academy of Sciences, Moscow.