Universality in ultradilute liquid Bose-Bose mixtures

We have studied dilute Bose-Bose mixtures of atoms with attractive interspecies and repulsive intraspecies interactions using quantum Monte Carlo methods at T=0. Using a number of models for interactions, we determine the range of validity of the universal equation of state of the symmetric liquid mixture as a function of two parameters: the s-wave scattering length and the effective range of the interaction potential. It is shown that the Lee-Huang-Yang correction is sufficient only for extremely dilute liquids with the additional restriction that the range of the potential is small enough. Based on the quantum Monte Carlo equation of state we develop a density functional which goes beyond the Lee-Huang-Yang term and use it together with the local density approximation to determine density profiles of realistic self-bound drops.Postprint (published version

Mean Value from Representation of Rational Number as Sum of Two Egyptian Fractions

For given positive integers $n$ and $a$, let $R(n;\,a)$ denote the number of positive integer solutions $(x,\,y)$ of the Diophantine equation $${a\over n}={1\over x}+{1\over y}.$$ Write $$S(N;\,a)=\sum_{\substack{n\leq N (n,\,a)=1}}R(n;\,a).$$ Recently Jingjing Huang and R. C. Vaughan proved that for $4\leq N$ and $a\leq 2N$, there is an asymptotic formula $$S(N;\,a)={3\over \pi^2a}\prod_{p|a}{p-1\over p+1}\cdot N(\log^2N+c_1(a) \log N+c_0(a))+\Delta(N;\,a).$$ In this paper, we shall get a more explicit expression with better error term for $c_0(a)$