Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems. VI. Asymptotics in the two-cluster region

Elliott H. Lieb, Barry Simon

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Abstract

We study the asymptotic behavior of the ground-state wave function of multiparticle quantum systems without statistics in that region of configuration space where the particles break up into two well-defined clusters very far apart. One example of our results is the following: consider a system of N particles moving in three dimensions with rotationally invariant two-body potentials which are bounded and have compact support. Let D = C1,C2 be a partition into two clusters so that H(C1) and H(C2) have discrete ground states η1 and η2 of energy ϵ1 and ϵ2. Suppose that Σ = ϵ1 + ϵ2 = inf σess(H) and that H has a discrete ground state ϑ of energy E. Let ζ1and ζ2 denote internal coordinates for the clusters C1 and c2 and let R be the difference of the centers of mass of the clusters. Let μ = M1M2/M1 + M2with Mi the mass of clusters Ci and define k by k2/2m = Σ-E. Then as R → a8 with ¦ζi¦ bounded, we prove that ϑ(ζ12, R) = cη(ζ1)η(ζ2)e−kRR−1(1+O(e−γR)) for some γ, c 〉 0. We prove weaker conclusions under weaker hypotheses, including results in the atomic case.

Original languageEnglish (US)
Pages (from-to)324-343
Number of pages20
JournalAdvances in Applied Mathematics
Volume1
Issue number3
DOIs
StatePublished - 1980

All Science Journal Classification (ASJC) codes

  • Applied Mathematics

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