Optimized lower bounds for N-body Hamiltonians
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Date
2006-05-23
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KheirEddine Boudjemaa
Abstract
Generalizing a method used previously for three-body, four-body and very
recently for five-body systems, we derive a lower bound for the ground state
energy of an N-body Hamiltonian, with arbitrary N. Because the expression of
the lower bound obtained in this way depends on a number of parameters, we
obtain in fact a family of lower bounds, a lower bound for each set of values of
these parameters. The best of these is of course obtained by maximizing over
these parameters and is correspondingly named optimized lower bound. The
set of values of the parameters corresponding to the optimized lower bound
satisfy a number of relations, named universal dynamical constraints, which
result from the application of a dynamical principle and are independent of the
particular form of the potential. For N = 3, 4, 5, they can be worked out in the
most general case. For N = 6 up, they can be worked out only for particular
mass configurations. Furthermore, the optimized lower bound proves to be
saturated in the harmonic oscillator case.