
 Friday, May 27, 2005
 Connection between the spectrum and the moments of the groundstate density in Ndimensional space
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Abstract. We show that a series of recurrent inequalities derived in N = 3 have the same formal expressions in any dimension N ≥ 2. They are derived from the multipole sum rules, and provide us with upper bounds for the moments of the groundstate density depending only on the lowest multipole excitation energy. These bounds are transformed into approximate recurrent relations by means of an empirical correction factor. The 1/r potential and the harmonic oscillator play a key role in establishing this factor, which is exact for these two potentials by construction. For a large class of potentials, we show that this factor tends to 1 as N → ∞. In such cases, at the largeN limit, the lowest state for each multipole excitation exhausts the sum rule. It thus acquires the characteristics of the onephonon excitation typical of the harmonic oscillator.
PACS numbers: 03.65.−w, 03.65.Ge, 02.30.Hq
