Bogoliubov excitation-spectrum conjecture for the Gross–Pitaevskii three-body Bose gas

Let $H_N^{\rm GP}$ be the Gross–Pitaevskii Hamiltonian, let $u_0$ be the minimizer of the Gross–Pitaevskii functional, and let $D$ and $b_{\mathcal M}(V)$ be defined by the Hessian of that functional. Define

$$E=\left(D^{1/2}\left(D+b_{\mathcal M}(V)u_0^4\right)D^{1/2}\right)^{1/2}.$$

Excitation-spectrum conjecture. The low-lying spectrum of $H_N^{\rm GP}$ is given by

$$\inf\sigma(H_N^{\rm GP})+\sum_{i\geqslant1}n_i e_i,$$

where $n_i\in\{0,1,2,\dots\}$ and $\{e_i\}_{i=1}^{\infty}$ are the positive eigenvalues of $E$. This is the expected Bogoliubov description of the excitations; establishing it for the three-body Gross–Pitaevskii regime remains open.