Jv Pule

Classical systems of infinitely many noninteracting particles

It is proved that the C*‐algebra of observables of an infinite classical system is isomorphic to the group algebra on the test function space D. The physical dynamical system consisting of infinitely many noninteracting particles is studied. A particular class of states, called the quasifree states, is exhibited and their properties are studied. Some results on the spectral properties of monoparticle evolutions are obtained. Finally we give explicitly a solution of the classical KMS condition for these evolutions.

Goldstone theorem for bose systems

In three or more dimensions (ν≥3) it is proved that if the correlations decay faster than |x|-(ν-20) then gauge symmetry breaking is excluded. In one and two dimensions (ν=1 or 2) the gauge symmetry is always preserved.

Linear response and relaxation in quantum lattice systems

For spin-lattice systems, the Kubo formula, expressing the relaxation function in terms of the linear response function, is found to be exact in the thermodynamic limit. In addition, analyticity properties are obtained.

The classical limit of quantum dissipative generators

For finite quantum systems the classical limit of the general dissipative generator of a semigroup of completely positive maps is obtained, yielding a generalized Fokker–Planck generator.

Peierls-Fröhlich instability and Kohn anomaly

A mathematical basis is given to the Peierls-Fröhlich instability and the Kohn anomaly. The techniques and ideas are based on the recently developed mathematical theory of quantum fluctuations and response theory. We prove that there exists a unique resonant one-mode interaction between electrons and phonons which is responsible for the Peierls-Fröhlich instability and the phase transition in the Mattis-Langer model. We prove also that the softening of this phonon mode at the critical temperature (Kohn anomaly) is a consequence of the critical slowing down of the dynamics of the lattice distortion fluctuations. It is the result of the linear dependence of two fluctuation operators corresponding to the frozen charge density wave and the distortion order parameter.

The solutions of the classical KMS-equation for infinitely many non-interacting particles

The complete solution of the classical KMS-equation for quasi-free evolutions is given under two different conditions.

Dissipative operators for infinite classical systems and equilibrium

Dissipative differential operators for infinite classical systems are characterized and used to obtain correlation inequalities for equilibrium states.

Dispersive properties and observables at infinity for classical kms systems

For infinite classical dynamical systems, satisfying the KMS condition, relations between asymptotic dispersive and cluster properties are proved. The local structure of the algebra of observables is explicitly characterized by the Poisson bracket commutant, and it is proved that the algebra of observables at infinity are constants of the motion.

Integral representations of the classical KMS-states for quasi-free evolutions

Abstract Integral representations are given for classical states of the type of the general solution of the KMS-equation for quasi-free evolution. The result is given for potentials tending to zero at infinity (Theorem 1) and for potentials increasing fast enough at infinity (Theorem 2).