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Valentin Zagrebnov

Publications and source records attributed to Valentin Zagrebnov.

29 records · Page 2Linked to original sources
arXiv2008

Numerical Range and Quasi-Sectorial Contractions

We apply a method developed by one of the authors, see \cite{Arl1}, to localize the numerical range of \textit{quasi-sectorial} contractions semigroups. Our main theorem establishes a relation between the numerical range of quasi-sectorial contraction semigroups $\{\exp(- t S)\}_{t\ge 0}$, and the maximal {sectorial} generators $S$. We also give a new prove of the rate $O(1/n)$ for the operator-norm Euler formula approximation: $\exp(- t S)=\lim\limits_{n\to \infty}(I+tS/n)^{-n}$, $t\ge 0$, for this class of semigroups.

math.FA↗
arXiv2008

Mean-Field Interacting Boson Random Point Fields in Weak Harmonic Traps

A model of the mean-field interacting boson gas trapped by a weak harmonic potential is considered by the \textit{boson random point fields} methods. We prove that in the Weak Harmonic Trap (WHT) limit there are two phases distinguished by the boson condensation and by a different behaviour of the local particle density. For chemical potentials less than a certain critical value, the resulting Random Point Field (RPF) coincides with the usual boson RPF, which corresponds to a non-interacting (ideal) boson gas. For the chemical potentials greater than the critical value, the boson RPF describes a divergent (local) density, which is due to \textit{localization} of the macroscopic number of condensed particles. Notice that it is this kind of transition that observed in experiments producing the Bose-Einstein Condensation in traps.

math-ph↗
arXiv2008

From Laplacian Transport to Dirichlet-to-Neumann (Gibbs) Semigroups

The paper gives a short account of some basic properties of \textit{Dirichlet-to-Neumann} operators $Λ_{γ,\partialΩ}$ including the corresponding semigroups motivated by the Laplacian transport in anisotropic media ($γ\neq I$) and by elliptic systems with dynamical boundary conditions. For illustration of these notions and the properties we use the explicitly constructed \textit{Lax semigroups}. We demonstrate that for a general smooth bounded convex domain $Ω\subset \mathbb{R}^d$ the corresponding {Dirichlet-to-Neumann} semigroup $\left\{U(t):= e^{-t Λ_{γ,\partialΩ}}\right\}_{t\geq0}$ in the Hilbert space $L^2(\partial Ω)$ belongs to the \textit{trace-norm} von Neumann-Schatten ideal for any $t>0$. This means that it is in fact an \textit{immediate Gibbs} semigroup. Recently Emamirad and Laadnani have constructed a \textit{Trotter-Kato-Chernoff} product-type approximating family $\left\{(V_{γ, \partialΩ}(t/n))^n \right\}_{n \geq 1}$ \textit{strongly} converging to the semigroup $U(t)$ for $n\to\infty$. We conclude the paper by discussion of a conjecture about convergence of the \textit{Emamirad-Laadnani approximantes} in the the {\textit{trace-norm}} topology.

math.FA↗
arXiv2006

First-order transitions for some generalized XY models

In this note we demonstrate the occurrence of first-order transitions in temperature for some recently introduced generalized XY models, and also point out the connection between them and annealed site-diluted (lattice-gas) continuous-spin models.

math-ph↗
arXiv2006

Bose-Einstein Condensation in the Luttinger-Sy Model

We present a rigorous study of the Bose-Einstein condensation in the Luttinger-Sy model. We prove the existence of the condensation in this one-dimensional model of the perfect boson gas placed in the Poisson random potential of singular point impurities. To tackle the off-diagonal long-range order we calculate explicitly the corresponding space-averaged one-body reduced density matrix. We show that mathematical mechanism of the Bose-Einstein condensation in this random model is similar to condensation in a one-dimensional nonrandom hierarchical model of scaled intervals. For the Luttinger-Sy model we prove the Kac-Luttinger conjecture, i.e., that this model manifests a type I BEC localized in a single "largest" interval of logarithmic size.

math-ph↗
arXiv2005

The Casimir effect for the Bose-Gas in Slabs

We study the Casimir effect for the perfect Bose-gase in the slab geometry for various boundary conditions. We show that the grand canonical potential per unit area at the bulk critical chemical potential $μ=0$ has the standard asymptotic form with universal Casimir terms.

cond-mat.stat-mech↗
arXiv2005

Lower Spectral Branches of a Particle Coupled to a Bose Field

The structure of the lower part (i.e. $ε$-away below the two-boson threshold) spectrum of Fröhlich's polaron Hamiltonian in the weak coupling regime is obtained in spatial dimension $d\geq 3$. It contains a single polaron branch defined for total momentum $p\in G^{(0)} $, where $G^{(0)}\subset {\mathbb R}^d$ is a bounded domain, and, for any $p\in {\mathbb R}^d$, a manifold of polaron + one-boson states with boson momentum $q$ in a bounded domain depending on $p$. The polaron becomes unstable and dissolves into the one boson manifold at the boundary of $G^{(0)}$. The dispersion laws and generalized eigenfunctions are calculated.

math-ph↗
arXiv2003

Magnetostriction Transition

We present a non mean-field model which undergoes a magnetostriction phase transition in the temperature. That is, the crystal becomes sharply contracted and magnetized once the temperature passes below the critical value

math-ph↗
arXiv2002

The equilibrium states for a model with two kinds of Bose condensation

We study the equilibrium Gibbs states for a Boson gas model, defined by Bru and Zagrebnov, which has two phase transitions of the Bose condensation type. The two phase transitions correspond to two distinct mechanisms by which these condensations can occur. The first (non-conventional) Bose condensation is mediated by a zero-mode interaction term in the Hamiltonian. The second is a transition due to saturation quite similar to the conventional Bose-Einstein (BE) condensation in the ideal Bose gas. Due to repulsive interaction in non-zero modes the model manifests a generalized type III, i.e., non-extensive BE condensation. Our main result is that, as in the ideal Bose gas, the conventional condensation is accompanied by a loss of strong equivalence of the canonical and grand canonical ensembles whereas the non-conventional one, due to the interaction, does not break the equivalence of ensembles. It is also interesting to note that the type of (generalized) condensate, I, II, or III (in the terminology of van den Berg, Lewis and Pule), has no effect on the equivalence of ensembles. These results are proved by computing the generating functional of the cyclic representation of the Canonical Commutation Relation (CCR) for the corresponding equilibrium Gibbs states.

math-ph↗