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Register a new userClassical $n$-body scattering with long-range potentials
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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.
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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 terms of the cardinality of the level sets of the potentials, which improves, in the perturbative regime, the result obtained by Goldstein and Schlag cite{gs2}. Our approach is a combination of Aubry duality, generalized Thouless formula and the regularity of the Lyapunov exponents of analytic quasi-periodic $GL(m,C)$ cocycles which is proved by quantitative almost reducibility method.
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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 quantum spin models that illustrate this scaling behaviour, and provide indications for its common occurrence by making use of Lieb-Robinson bounds. I argue that these findings may help in understanding the extraordinarily short equilibration timescales predicted by typicality techniques.
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