Jan Dereziński

Perturbation theory of W*-dynamics, Liouvilleans and KMS states

Given a W*-algebra

Fermi Golden Rule and Open Quantum Systems

{\mathfrak M}

A new proof of the propagation theorem for

with a W*-dynamics τ, we prove the existence of the perturbed W*-dynamics for a large class of unbounded perturbations. We compute its Liouvillean. If τ has a β-KMS state, and the perturbation satisfies some mild assumptions related to the Golden–Thompson inequality, we prove the existence of a β-KMS state for the perturbed W*-dynamics. These results extend the well known constructions due to Araki valid for bounded perturbations.

N

2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Level Shift Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 LSO for C∗ 0 -dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 LSO for W ∗-dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 LSO in Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.6 The choice of the projection P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.7 Three kinds of the Fermi Golden Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

-body quantum systems

A new proof of I. Sigals and A. Soffers propagation theorem is given. This theorem describes a large class of operators which are Kato-smooth with respect to anN-body Schrödinger operator.

Exactly Solvable Schrödinger Operators

We systematically describe and classify one-dimensional Schrödinger equations that can be solved in terms of hypergeometric type functions. Beside the well-known families, we explicitly describe two new classes of exactly solvable Schrödinger equations that can be reduced to the Hermite equation.

Fluctuations of Quantum Currents and Unravelings of Master Equations

The very notion of a current fluctuation is problematic in the quantum context. We study that problem in the context of nonequilibrium statistical mechanics, both in a microscopic setup and in a Markovian model. Our answer is based on a rigorous result that relates the weak coupling limit of fluctuations of reservoir observables under a global unitary evolution with the statistics of the so-called quantum trajectories. These quantum trajectories are frequently considered in the context of quantum optics, but they remain useful for more general nonequilibrium systems. In contrast with the approaches found in the literature, we do not assume that the system is continuously monitored. Instead, our starting point is a relatively realistic unitary dynamics of the full system

Bogoliubov Hamiltonians and one-parameter groups of Bogoliubov transformations

On the bosonic Fock space, a family of Bogoliubov automorphisms corresponding to a strongly continuous one-parameter group of symplectic maps (R(t))t∊R is considered. We give conditions that guarantee it to be implemented by a strongly continuous one-parameter group U(t) of unitary operators. The generator of such U(t) will be called a Bogoliubov Hamiltonian. Given (R(t))t∊R, a Bogoliubov Hamiltonian is defined up to an additive constant. We introduce two kinds of Bogoliubov Hamiltonians: type I, characterized by vanishing of the expectation value at the vacuum, and type II, characterized by the fact that its infimum equals zero. We give conditions so that they are well defined. We show that there exist cases when only HI is well defined, even though the classical Hamiltonian is positive (which may be interpreted as a kind of an infrared catastrophe), and when only HII is well defined (which means that one needs to introduce an infinite counterterm in the formula for the Hamiltonian).

Introduction to Representations of the Canonical Commutation and Anticommutation Relations

Lecture notes of a minicourse given at the Summer School on Large Coulomb Systems - QED in Nordfjordeid, 2003, devoted to representations of the CCR and CAR. Quasifree states, the Araki-Woods and Araki-Wyss representations, and the lattice of von Neumenn algebras in a bosonic/fermionic Fock space are discussed in detail.

Excitation Spectrum of Interacting Bosons in the Mean-Field Infinite-Volume Limit

We consider homogeneous Bose gas in a large cubic box with periodic boundary conditions, at zero temperature. We analyze its excitation spectrum in a certain kind of a mean-field infinite-volume limit. We prove that under appropriate conditions the excitation spectrum has the form predicted by the Bogoliubov approximation. Our result can be viewed as an extension of the result of Seiringer (Commun. Math. Phys. 306:565–578, 2011) to large volumes.