Quantization in the neighborhood of classical solutions in the $\boldsymbol N$ particle problem and superfluidity

We have considered the system of $N$ similar interacting bosons in the external field. Hamiltonian of the system is
$$ \widehat H_N=\sum_{i=1}^{N}\bigl(-\Delta_i/2+U(x_i)\bigr)+\varepsilon\sum_{1\le i<j\le N} V(x_i-x_j). $$
We have found asimptotical series of eigenvalues and eigenfunctions of $\widehat H_N$ if $N\to\infty$, $\varepsilon\to0$, $\varepsilon N\to\alpha=\text{const}$. These series correspond with stable solutions of Hartree equation
$$ \bigl(-\Delta/2+U(x)\bigr) f(x)+\alpha\int dy\,V(x-y)\,|f(y)|^2f(x)=\Omega f(x). $$
If $U=0$, $f(x)=\text{const}\cdot\exp(ipx)$ then out result is in agreement with Bogolubov's work about superfluidity. Phenomena analogous with superfluidity arises in other cases, too.