Energy inequalities({N}{2})−1EN≤({K}{2})−1EK,2≤K<N, relating two systems of identical bosons

Abstract

Suppose ${E}_{N}$ is the lowest energy of a translation-invariant system of $N$ identical bosons which interact by attractive central pair potentials and obey nonrelativistic quantum mechanics. Sufficient conditions are prescribed on the potential to prove that ${E}_{N}$ is bounded above by $({N}{2}){({K}{2})}^{\ensuremath{-}1}{E}_{K}$, where $N\ensuremath{\ge}\ensuremath{\mu}K$. The $\ensuremath{\mu}$ numbers are specified for various classes of potential shape. In cases where $\ensuremath{\mu}\ensuremath{\le}\frac{3}{2}$, it follows that ${E}_{N}\ensuremath{\le}\frac{1}{2}N(N\ensuremath{-}1){E}_{2}$ for $N\ensuremath{\ge}3$.