O'Connor's approach to spatial exponential decay of eigenfunctions for multiparticle Schrodinger Hamiltonians is developed from the point of view of analytic perturbations with respect to transformation groups.This framework allows an improvement of his results in some directions; in particular if interactions are dilation analytic, exponential fall-off is shown to hold for any bound-state wave-function corresponding to an eigenvalue distinct from thresholds; it is shown that the exponential decay rate depends on the distance from the bound-state energy to the nearest threshold. Applications include non existence of positive energy bound-states.

A time dependent scattering theory for a quantum mechanical particle moving in an infinite, three dimensional crystal with impurity is given. It is shown that the Hamiltonian for the particle in the crystal without impurity has only absolutely continuous spectrum. The domain of the resulting wave operators is therefore the entire Hubert space.

We continue the study of a model for heat conduction [6] consisting of a chain of non-linear oscillators coupled to two Hamiltonian heat reservoirs at different temperatures. We establish existence of a Liapunov function for the chain dynamics and use it to show exponentially fast convergence of the dynamics to a unique stationary state. Ingredients of the proof are the reduction of the infinite dimensional dynamics to a finite-dimensional stochastic process as well as a bound on the propagation of energy in chains of anharmonic oscillators. 1

Dedicated to the memory of R.L. Dobrushin and P.W. Kasteleyn, two founders of modern statistical mechanics. AbstractThe polaron has been of interest in condensed matter theory and field theory for about half a century, especially the limit of large coupling constant, α. It was not until 1983, however, that a proof of the asymptotic formula for the ground state energy was finally given by using difficult arguments involving the large deviation theory of path integrals. Here we derive the same asymptotic result, E 0 ∼ −Cα 2 , and with explicit error bounds, by simple, rigorous methods applied directly to the Hamiltonian. Our method is easily generalizable to other settings, e.g., the excitonic and magnetic polarons.The polaron Hamiltonian of Fröhlich [1] is a model for the Coulomb interaction of one or more electrons with the quantized phonons of an ionic crystal. In the course of time it was also seen to be an interesting field theory model of non-relativistic particles interacting with a scalar boson field, and it was widely studied [2,3,4,5,6,7,8,9] in both contexts.

We prove the Gallavotti-Cohen fluctuation theorem for a model of heat conduction through a chain of anharmonic oscillators coupled to two Hamiltonian reservoirs at different temperatures.