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We consider the scattering of $n$ classical particles interacting via pair potentials, under the assumption that each pair potential is long-range, i.e. being of order ${cal O}(r^{-alpha})$ for some $alpha >0$. We define and focus on the free region, the set of states leading to well-defined and well-separated final states at infinity. As a first step, we prove the existence of an explicit, global surface of section for the free region. This surface of section is key to proving the smoothness of the map sending a point to its final state and to establishing a forward conjugacy between the $n$-body dynamics and free dynamics.
We prove the Holder continuity of the integrated density of states for a class of quasi-periodic long-range operators on $ell^2(Z^d)$ with large trigonometric polynomial potentials and Diophantine frequencies. Moreover, we give the Holder exponent in
We introduce an algebraic method to study local stability in the Newtonian $n$-body problem when certain symmetries are present. We use representation theory of groups to simplify the calculations of certain eigenvalue problems. The method should be
We apply the renormalisation-group to two-body scattering by a combination of known long-range and unknown short-range forces. A crucial feature is that the low-energy effective theory is regulated by applying a cut-off in the basis of distorted wave
We study the direct and inverse scattering problem for the one-dimensional Schrodinger equation with steplike potentials. We give necessary and sufficient conditions for the scattering data to correspond to a potential with prescribed smoothness and
Long-range interacting many-body systems exhibit a number of peculiar and intriguing properties. One of those is the scaling of relaxation times with the number $N$ of particles in a system. In this paper I give a survey of results on long-range quan