Existence conjecture for Gross–Pitaevskii states with a linear counterpart

Let $U_0$ be the nonlinear interaction coefficient, let $\mu$ be the chemical potential, and let $(\mathcal{E}_n,\phi_n)$ be an eigenvalue and eigenfunction of the associated linear problem. A stationary state with a linear counterpart has the form $\psi\simeq\sqrt{N(\mu)}\phi_n$, where $N(\mu)$ is its particle number. Existence conjecture for states with a linear counterpart. For $U_0>0$ $(U_0<0)$, solutions with linear limit $\psi\simeq\sqrt{N(\mu)}\phi_n$ exist only if $\mu>\mathcal{E}_n$ $(\mu<\mathcal{E}_n)$. Moreover, $N(\mu)\to0$ for $\mu\to\mathcal{E}_n$. This conjecture concerns the bifurcation of nonlinear stationary Gross–Pitaevskii states from eigenfunctions of the associated linear problem as the particle number tends to zero; its resolution is not specified in the source.