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Proceedings of a Workshop Held at Aarhus, Denmark 15 May - 1 August 1991

*V*on

**R**

^{ n }and the spectrum of the corresponding self-adjoint Hamiltonian

*H*= −Δ +

*V*on

*L*

^{2}(

**R**

^{ n }). More specifically, for a fixed energy interval

*I*,we want to examine the effects of strong local fluctuations of

*V*near energies in

*I*on the spectrum...

*H*= −Δ +

*V*acting on

*L*

^{2}(

**R**

^{ n }), where

*V*is a bounded real-valued function of class

*C*

^{ l }on

**R**

^{ n }\ {0} with

*n*≥ 2 that satisfies 1.1 $\begin{array}{cc}{\partial}_{x}^{\alpha}V\left(x\right)=O\left({\left|x\right|}^{-\left|\alpha \right|}\right),& as\text{\hspace{0.05em}}\left|x\right|\to \infty ,\left|\alpha \right|\le \end{array}$...

*N*-body quantum systems with potentials that decay like

*x*

^{−μ }where $\mu >\sqrt{3}-1$]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\mu > \sqrt 3 - 1$$ .

*N*—particle Hamiltonians with long range pair potentials decaying at infinity like <

*x*>

^{−μ }for

*μ*> 1/2 are asymptotically clustering at all non-threshold energies.

*N*quantized electrons at positions

*x*

_{i}, and

*M*nuclei of charges

*Z*= (

*Z*

_{1},...,

*Z*

_{M}) fixed at positions

*y*= (

*y*

_{l},...,

*y*

_{M}). The Schrödinger Hamiltonian of such a system is given by ${H}_{Z,H}=\sum _{i=1}^{N}\left(-{\Delta}_{{x}_{i}}+{V}_{Coulomb}\left({x}_{i}\right)\right)+\frac{1}{2}\sum _{i\ne j}\frac{1}{\left|{x}_{i}-{x}_{j}\right|}$]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${H_{Z,H}}...

*Z*becomes infinite with the number of nuclei and their charge ratios fixed. It is shown that, in the Born-Oppenheimer approximation, if such a system has a stable bound state then it is asymptotically neutral in the sense that it satisfies the inequality |

*Z*−

*N*| <

*C*

_{1}Z

^{1−ε }where

*N*denotes the number of electrons,...

*r*→ ∞ of the difference of the sojourn times of a scattering state and of the associated free state in a ‘fuzzy’ ball of radius

*r*in

*ℝ*. The potential

*W*is assumed to be smooth and behave like |

*|*

__x__^{−α }(

*α*> 1) at infinity. For earlier studies on this...

*N*-cluster amplitudes under a short range condition on the potential and in addition under a discreteness assumption on the 2-cluster channel energies. This gives a rather complete picture for

*N*= 3 while a number...

*H*

_{ ћ }= −

*ћ*

^{2}

*∂*

_{ φ }

^{2}/2+U(

*φ*) be a Schrödinger operator acting in

*L*

^{2}(T) with T an

*ℓ*-dimensional torus and

*U*an analytic periodic function on T. Approximate semiclassical expansions for the eigenfunctions and eigenvalues of

*H*

_{ ћ }are developed which are asymptotic...

*N*-particle quantum systems. The estimates obtained allow us to give an elementary proof of asymptotic completeness for such systems in the framework of the theory of smooth perturbations.