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 $\Lambda_{\gamma,\partial\Omega}$ including the corresponding semigroups motivated by the Laplacian transport in anisotropic media ($\gamma \neq I$) 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 $\Omega \subset \mathbb{R}^d$ the corresponding {Dirichlet-to-Neumann} semigroup $\left\{U(t):= e^{-t \Lambda_{\gamma,\partial\Omega}}\right\}_{t\geq0}$ in the Hilbert space $L^2(\partial \Omega)$ 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 $\left\{(V_{\gamma, \partial\Omega}(t/n))^n \right\}_{n \geq 1}$ \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

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

A Dynamics Driven by Repeated Harmonic Perturbations

We propose an exactly soluble W*-dynamical system generated by repeated harmonic perturbations of the one-mode quantum oscillator. In the present paper we deal with the case of isolated system. Although dynamics is Hamiltonian and quasi-free, it produces relaxation of initial state of the system to the steady state in the large-time limit. The relaxation is accompanied by the entropy production and we found explicitly the rate for it. Besides, we study evolution of subsystems to elucidate their eventual correlations and convergence to equilibrium state. Finally we prove a universality of the dynamics driven by repeated harmonic perturbations in a certain short-time interaction limit

Quasi-Sectorial Contractions

We revise the notion of the quasi-sectorial contractions. Our main theorem establishes a relation between semigroups of quasi-sectorial contractions and a class of m-sectorial generators. We discuss a relevance of this kind of contractions to the theory of operator-norm approximations of strongly continuous semigroups

Dynamics of an Open System for Repeated Harmonic Perturbation

We use the Kossakowski-Lindblad-Davies formalism to consider an open system defined as the Markovian extension of one-mode quantum oscillator S, perturbed by a piecewise stationary harmonic interaction with a chain of oscillators C. The long-time asymptotic behaviour of various subsystems of S+C are obtained in the framework of the dual W-dynamical system approach

The second critical point for the Perfect Bose gas in quasi-one-dimensional traps

We present a new model of quasi-one-dimensional trap with some unknown physical predictions about a second transition, including about a change in fractions of condensed coherence lengths due to the existence of a second critical temperature Tm < Tc. If this physical model is acceptable, we want to challenge experimental physicists in this regard