Collision operator in the physical representation, the friedrichs model

Ilya Prigogine, Evgueni Karpov, Tomio Petrosky +1 more

A.P. Grécos, I. Prigogine

We consider the dissipative properties of large quantum systems from the point of view of kinetic theory. The existence of a nontrivial collision operator imposes restrictions on the possible collisional invariants of the system. We consider a model in which a discrete level is coupled to a set of quantum states and which, in the limit of a large "volume," becomes the Friedrichs model. Because of its simplicity this model allows a direct calculation of the collision operator as well as of related operators and the constants of the motion. For a degenerate spectrum the calculations become more involved but the conclusions remain simple. The special role played by the invariants that are functions of the Hamiltonion is shown to be a direct consequence of the existence of a nonvanishing collision operator. For a class of observables we obtain ergodic behavior, and this reformulation of the ergodic problem may be used in statistical mechanics to study the ergodicity of large quantum systems containing a small physical parameter such as the coupling constant or the concentration.

Pierre M. V. Résibois

Mario Castagnino, Graciela Domenech, Marcelo Leonardo Levinas +1 more

Applying the doublet representation we analyze the solutions of a Hamiltonian system which has eigenstates with complex eigenvalues. The example of the Friedrichs model allows us to show how the appearance of solutions with non-Hilbert initial conditions is linked to the energy degeneration of the Hamiltonian spectrum. We discuss the difficulties of giving a physical meaning to the growing or decaying non-Hilbert solutions. We also suggest a way to circumvent the problem of the anomalous probabilities related to both complex energy eigenvalues and degeneration of the spectrum.

Lawrence Paul Horwitz

A brief summary of the mathematical structure of the Lee-Friedrichs model is given, including the natural extension of its Hamiltonian to a Gel'fand triple, containing a generalized eigenstate for the complex pole. Contains references to some physical applications and relativistic generalizations.

Pierre M. V. Résibois