Open Quantum Systems I

1. Front Matter

   Pages i-xvii

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2. Introduction to the Theory of Linear Operators

   • Alain Joye

   Pages 1-40

3. Introduction to Quantum Statistical Mechanics

   • Alain Joye

   Pages 41-67

4. Elements of Operator Algebras and Modular Theory

   • Stéphane Attal

   Pages 69-105

5. Quantum Dynamical Systems

   • Claude-Alain Pillet

   Pages 107-182

6. The Ideal Quantum Gas

   • Marco Merkli

   Pages 183-233

7. Topics in Spectral Theory

   • Vojkan Jakšić

   Pages 235-312

8. Back Matter

   Pages 313-333

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This is the ?rst in a series of three volumes dedicated to the lecture notes of the Summer School ”Open Quantum Systems” which took place at the Institut Fourier in Grenoble from June 16th to July 4th 2003. The contributions presented in these volumes are revised and expanded versions of the notes provided to the students during the School. Closed vs. Open Systems By de?nition, the time evolution of a closed physical systemS is deterministic. It is usually described by a differential equation x ? = X(x ) on the manifold M of t t possible con?gurations of the system. If the initial con?guration x ? M is known 0 then the solution of the corresponding initial value problem yields the con?guration x at any future time t. This applies to classical as well as to quantum systems. In the t classical case M is the phase space of the system and x describes the positions and t velocities of the various components (or degrees of freedom) ofS at time t. Inthe quantum case, according to the orthodox interpretation of quantum mechanics, M is a Hilbert space and x a unit vector – the wave function – describing the quantum t state of the system at time t. In both cases the knowledge of the state x allows t to predict the result of any measurement made onS at time t.

• Markov process
• Quantum dynamical systems
• algebra
• dynamical systems
• equation
• mathematical physics
• non-equilibrium statistical mechanics
• operator algebras
• quantum noises
• spectral theory
• stochastic differential equation

• Institut Camille Jordan, Université Claude Bernard Lyon 1, Villeurbanne Cedex, France

  Stéphane Attal

• Institut Fourier, Université de Grenoble 1, Saint-Martin d'Hères Cedex, France

  Alain Joye

• CPT-CNRS, UMR 6207, Université du Sud Toulon-Var, La Garde Cedex, France

  Claude-Alain Pillet