25 research outputs found
An alternative to Plemelj-Smithies formulas on infinite determinants
An alternative to Plemelj - Smithies formulas for the p -regularized
quantities and is presented which generalizes
previous expressions with due to Grothendieck and Fredholm. It is also
presented global upper bounds for these quantities. In particular we prove that
|d^{(p)}(K)| \le e^{\kappa \n K \n_p^p} holds with for which improves previous estimate yielding .Comment: CYCLER Paper 93Jan00
On the Mayer series of two-dimensional Yukawa gas at inverse temperature in the interval of collapse
A Theorem on the minimal specific energy for a system with \pm 1 charged
particles interacting through the Yukawa pair potential v is proved which may
stated as follows. Let v be represented by scale mixtures of d-dimensional
Euclid's hat (cutoff at short scale distances) with d\geq 2. For any even
number of particles n, the interacting energy U_{n} divided by n, attains an
n-independent minimum at a configuration with zero net charge and particle
positions collapsed altogether to a point. For any odd number of particles n,
the ratio U_{n}/(n-1) attains its minimum value, the same of the previous
cases, at the configuration with \pm 1 net charge and particle positions
collapsed to a point. This Theorem is then used to resolve an obstructive
remark of an unpublished paper (Remark 7.5 of \cite{Guidi-Marchetti}) which,
whether the standard decomposition of the Yukawa potential into scales were
adopted, would impede a direct proof of the convergence of the Mayer series of
the two-dimensional Yukawa gas for the inverse temperature in the whole
interval [4\pi ,8\pi ) of collapse. In the present paper, it is proven
convergence up to the second threshold 6\pi and its given explanations on the
mechanism that allow it to be extend up to 8\pi . The paper distinguishes the
matters concerning stability from those related to convergence of the Mayer
series. In respect to the latter the paper dedicates to the Cauchy majorante
method applied to the density function of Yukawa gas in the interval of
collapses. It also dedicates to the proof of the main Theorem and estimates of
the modified Bessel functions of second kind involved in both representations
of two-dimensional Yukawa potential: standard and scale mixture of the Euclid's
hat function.Comment: 49 pages, 7 figure
The Falicov-Kimball Model with Long--Range Hopping Matrices
The ground state nature of the Falicov-Kimball model with unconstrained
hopping of electrons is investigated. We solve the eigenvalue problem in a
pedagogical manner and give a complete account of the ground state energy both
as a function of the number of electrons and nuclei and as a function of the
total number of particles for any value of interaction U. We also study the
energy gap and show the existence of a phase transition characterized by the
absence of gap at the half--filled band for U<0. The model in consideration was
proposed and solved by Farkasovsky for finite lattices and repulsive on-site
interaction U>0. Contrary to his proposal we conveniently scale the hopping
matrix to guarantee the existence of the thermodynamic limit. We also solve
this model with bipartite unconstrained hopping matrices in order to compare
with the Kennedy--Lieb variational analysis.Comment: 23 pages, Late
On the Virial Series for a Gas of Particles with Uniformly Repulsive Pairwise Interaction
The pressure of a gas of particles with a uniformly repulsive pair
interaction in a finite container is shown to satisfy (exactly as a formal
object) a "viscous" Hamilton-Jacobi (H-J) equation whose solution in power
series is recursively given by the variation of constants formula. We
investigate the solution of the H-J and of its Legendre transform equation by
the Cauchy-Majorant method and provide a lower bound to the radius of
convergence on the virial series of the gas which goes beyond the threshold
established by Lagrange's inversion formula. A comparison between the behavior
of the Mayer and virial coefficients for different regimes of repulsion
intensity is also provided.Comment: Preliminary version; 62 pages, 9 figure
Rakib-Sivashinsky and Michelson-Sivashinsky Equations for Upward Propagating Flames: A Comparison Analysis
We establish a comparison between Rakib--Sivashinsky and
Michelson-Sivashinsky quasilinear parabolic differential equations governing
the weak thermal limit of upward flame front propagating in a channel. For the
former equation, we give a complete description of all steady solutions and
present their local and global stability analysis. For the latter,
multi-coalescent unstable steady solutions are introduced and shown to be
exponentially more numerous than the previous known coalescente solutions. This
fact is argued to be responsible for the disagreement of the observed dynamics
in numerical experiments with the exact (linear) stability analysis and also
gives the ingredients to describe the quasi-stable behavior of parabolic
steadily propagating flame with centered tip.Comment: 17 pages, 3 figure
Renormalization Group Flow of the Two-Dimensional Hierarchical Coulomb Gas
We consider a quasilinear parabolic differential equation associated with the
renormalization group transformation of the two-dimensional hierarchical
Coulomb system in the limit as the size of the block L goes to 1. We show that
the initial value problem is well defined in a suitable function space and the
solution converges, as t goes to infinity, to one of the countably infinite
equilibrium solutions. The nontrivial equilibrium solution bifurcates from the
trivial one. These solutions are fully described and we provide a complete
analysis of their local and global stability for all values of inverse
temperature. Gallavotti and Nicolo's conjecture on infinite sequence of
``phases transitions'' is also addressed. Our results rule out an intermediate
phase between the plasma and the Kosterlitz-Thouless phases, at least in the
hierarchical model we consider.Comment: 34pages,2figures, to appear in CM
Hierarchical Spherical Model as a Viscosity Limit of Corresponding O(N)Heisenberg Model
The O(N) Heisenberg and spherical models with interaction given by the long
range hierarchical Laplacean are investigated. Two classical results are
adapted. The Kac--Thompson solution [KT] of the spherical model, which holds
for spacially homogeneous interaction, is firstly extended to hierarchical
model whose interaction fails to be translation invariant. Then, the
convergence proof of O(N) Heisenberg to the spherical model by Kunz and Zumbach
[KZ] is extended to the long range hierarchical interaction. We also examine
whether these results can be carried over as the size of the hierarchical block
L goes to 1. These extensions are considered a preliminary study prior the
investigation of the model by renormalization group given in [MCG] where
central limit theorems for the spherical (N=\infty) model on the local
potential approximation (L\downarrow 1) are then established from an explicit
solution of the associate nonlinear first order partial differential equation
Combinatorial Solution of One-Dimensional Quantum Systems
We give a self-contained exposition of the combinatorial solution of quantum
mechanical systems of coupled spins on a one-dimensional lattice. Using Trotter
formula, we write the partition function as a generating function of a spanning
subgraph of a two-dimensional lattice and solve the combinatorial problem by
the method of Pfaffians provided the weights satisfy the so-called free fermion
condition. The free energy and the ground state energy as a function of the
inverse temperature, couplings J and magnetic fields h, for the XY model in a
transverse field with period p=1 and 2, is then obtained.Comment: Number of figures: 9 (eps
Singular perturbation of nonlinear systems with regular singularity
We extend Balser-Kostov method of studying summability properties of a
singularly perturbed inhomogeneous linear system with regular singularity at
origin to nonlinear systems of the form \varepsilon zf^{\prime} =
F(\varepsilon,z,f) with F a \mathbb{C}^{\nu}-valued function, holomorphic in a
polydisc \bar{D}_{\rho}\times \bar{D}_{\rho}\times \bar{D}_{\rho}^{\nu}. We
show that its unique formal solution in power series of \varepsilon, whose
coefficients are holomorphic functions of z, is 1-summable under a Siegal-type
condition on the eigenvalues of F_{f}(0,0,0). The estimates employed resemble
the ones used in KAM theorem. A simple Lemma is developed to tame convolutions
that appears in the power series expansion of nonlinear equations.Comment: 18 page
Conformal Deformation from Normal to Hermitian Random Matrix Ensembles
We investigate the eigenvalues statistics of ensembles of normal random
matrices when their order N tends to infinite. In the model the eigenvalues
have uniform density within a region determined by a simple analytic polynomial
curve. We study the conformal deformations of normal random ensembles to
Hermitian random ensembles and give sufficient conditions for the latter to be
a Wigner ensemble.Comment: 23 page