Binding conjecture for the nonlinear Schrödinger equation with orthonormal functions

Let $d\geq1$, let $N\geq2$, and define

$$J(N)=\inf\left\{\mathcal E(u_1,\ldots,u_N):u_1,\ldots,u_N\in H^1(\mathbb R^d,\mathbb C),\ \langle u_j,u_k\rangle_{L^2}=\delta_{jk}\right\},$$

where

$$\mathcal E(u_1,\ldots,u_N)=\sum_{n=1}^N\int_{\mathbb R^d}|\nabla u_n(x)|^2\,\mathrm dx-\frac1p\int_{\mathbb R^d}\left(\sum_{n=1}^N|u_n(x)|^2\right)^p\,\mathrm dx.$$

Binding conjecture. For every $N\geq2$ and every $1<p<\min\left(2,1+\frac{2}{d}\right)$, the binding inequalities

$$J(N)<J(N-K)+J(K),\qquad K=1,\ldots,N-1,$$

hold. In particular, $J(N)$ admits a minimiser, which is a ground state for the fermionic nonlinear Schrödinger equation.

These inequalities are the mechanism used to obtain ground states under the orthonormality constraint. The paper proves them only in selected regimes, including an infinite sequence of particle numbers, so the assertion for every $N$ in the stated range remains open.