I have a $N$-body Sch\"odinger operator of the following form \begin{equation} H := \sum_{j=1}^{N} (-\Delta_{x_j}+U_{trap}(x_j))+\sum_{i<j} V(x_i-x_j), \end{equation} where $U_{trap}$ is a positive function that traps, namely \begin{equation} \lim_{|x|\to+\infty}U(x)=+\infty, \end{equation} and $V$ has the property \begin{equation} V^2(x)\le C(1-\Delta_x). \end{equation} The Hamiltonian $H$ acts on the bosonic Hilbert space $L^2(\mathbb{R}^3)^{\otimes_{\operatorname{sym}}N}$.
I need to show that $H$ has compact resolvent, and I know that the kinetic part has compact resolvent. Could you help me in showing that the interaction between particles does not spoil the property? Thank you