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

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

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

Random point field approach to analysis of anisotropic Bose-Einstein condensations

Position distributions of constituent particles of the perfect Bose-gas trapped in exponentially and polynomially anisotropic boxes are investigated by means of the boson random point fields (processes) and by the spatial random distribution of particle density. Our results include the case of \textit{generalised} Bose-Einstein Condensation. For exponentially anisotropic quasi two-dimensional system (SLAB), we obtain \textit{three} qualitatively different particle density distributions. They correspond to the \textit{normal} phase, the quasi-condensate phase (type III generalised condensation) and to the phase when the type III and the type I Bose condensations co-exist. An interesting feature is manifested by the type II generalised condensation in one-directional polynomially anisotropic system (BEAM). In this case the particle density distribution rests truly random even in the \textit{macroscopic} scaling limit

On ergodic states, spontaneous symmetry breaking and the Bogoliubov quasi-averages

It is shown that Bogoliubov quasi-averages select the pure or ergodic states in the ergodic decomposition of the thermal (Gibbs) state. Our examples include quantum spin systems and many-body boson systems. As a consequence, we elucidate the problem of equivalence between Bose-Einstein condensation and the quasi-average spontaneous symmetry breaking (SSB) discussed for continuous boson systems. The multi-mode extended van den Berg-Lewis-Pul\'{e} condensation of type III demonstrates that the only physically reliable quantities are those that defined by Bogoliubov quasi-averages

Lieb-Robinson Bounds and Existence of the Thermodynamic Limit for a Class of Irreversible Quantum Dynamics

We prove Lieb-Robinson bounds and the existence of the thermodynamic limit for a general class of irreversible dynamics for quantum lattice systems with time-dependent generators that satisfy a suitable decay condition in space.Comment: Added 3 references and comments after Theorem 2; corrected typo

Remarks on the operator-norm convergence of the Trotter product formula

We revise the operator-norm convergence of the Trotter product formula for a pair {A,B} of generators of semigroups on a Banach space. Operator-norm convergence holds true if the dominating operator A generates a holomorphic contraction semigroup and B is a A-infinitesimally small generator of a contraction semigroup, in particular, if B is a bounded operator. Inspired by studies of evolution semigroups it is shown in the present paper that the operator-norm convergence generally fails even for bounded operators B if A is not a holomorphic generator. Moreover, it is shown that operator norm convergence of the Trotter product formula can be arbitrary slow.Comment: 12 page