Bounds for the phonon-roton dispersion in superfluid 4He

Abstract

The sum rule approach is used to derive upper bounds for the dispersion law $\ddot{I}{‰}_0(q)$ of the elementary excitations of a Bose superfluid. Bounds are explicitly calculated for the phonon-roton dispersion in superfluid ${}^4$He, both at equilibrium ($\ddot{I}=0.02186$ à ${}^{-3}$) and close to freezing ($\ddot{I}=0.02622$ à ${}^{-3}$). The bound $\ddot{I}{‰}_0(q)\le 2S(q)\mid \ddot{I}‡(q){\mid }^{-1}$, where $S(q)$ and $\ddot{I}‡(q)$ are the static structure factor and density response respectively, is calculated microscopically for several values of the wavevector $q$. The results provide a significant improvement with respect to the Feynman approximation $\ddot{I}{‰}_F(q)=q^2(2mS(q){)}^{-1}$. A further, stronger bound, requiring the additional knowledge of the current correlation function is also investigated. New results for the current correlation function are presented.

13 pages, REVTEX, 7 figures available upon request or at http://anubis.science.unitn.it/~dalfovo/papers/papers.html