The ground state energy of a Bose gas with Coulomb interaction

Abstract

LetH _N be the 2N particle Hamiltonian

$$\begin{array}{*{20}c} {H_N = \sum\limits_{i = 1}^{2N} {( - \Delta _\iota ) + \sum\limits_{i< j = 1}^N {\left| {x_i - x_j } \right|^{ - 1} + } \sum\limits_{i< j = 1}^N {\left| {x_{i + N} - x_{j + N} } \right|^{ - 1} } } } \\ { - \sum\limits_{i,j< j = 1}^N {\left| {x_i - x_{j + N} } \right|^{ - 1} ,} } \\ \end{array} $$

whereΔ _i is the Laplacian in the variablex _i ∈ℝ³, 1≦i≦2N. The operatorH _N is assumed to act on wave functionsΨ(x ₁, ...,x _N ;x _N+1, ...,x _2N ) which are symmetric in the variables (x ₁, ...,x _N ) and (x _N+1, ...,x _2N ). SupposeΨ is supported in a setΛ ^2N, whereΛ is a cube in ℝ³. It is shown that if a normalized wave functionΨ can be written as a product of two wave functions

$$\psi (x_1 ,...,x_N ;x_{N + 1} ,...,x_{2N} ) = \psi _1 (x_2 ,...,x_N )\psi _2 (x_{N + 1} ,...,x_{2N} ),$$

and the density of particles inΛ is constant, then 〈Ψ|H _N |Ψ〉≧−CN ^7/5 for some universal constantC.