Cubic Nonlinear Schrödinger Systems and Orthonormal Fermionic Ground States in $\mathbb{R}$

Abstract

We study the one-dimensional nonlinear Schrödinger system \[ -u_i''-2\Big(\sum_{k=1}^N u_k^2\Big)^{p-1}u_i=μ_i u_i \quad \mbox{in } \ \mathbb R,\ \ i=1,2,\cdots,N, \tag{S} \] where $u=(u_1,\cdots ,u_N)\in H^1(\mathbb{R}{)}^N$, ${\mu }_1\le {\mu }_2\le \cdots \le {\mu }_N<0$, $1<p<3$ and $N\ge 2$ are arbitrary. For the cubic case where $p=2$ and $N\ge 2$, we establish the following structural results of the system $(S)$: (i) all nontrivial solutions can be classified completely; (ii) the linearized operator at any nontrivial solution is non-degenerate; (iii) exact $L^2$-mass formulas of all nontrivial solutions are derived in terms of distinct Lagrange multipliers ${\mu }_i$. As the applications of the above structural results, we first show that the system $(S)$ with $p=2$ does not have any orthonormal solution for any $N\ge 2$. We then investigate a fermionic variational problem $J_N(p)$ with $N\ge 2$, whose minimizers correspond to ground states of the system $(S)$ under orthonormal constraints. More precisely, we prove that there exists a sufficiently small ${\delta }_N>0$ such that $J_N(p)$ is attained for any $p\in (2-{\delta }_N,2)$, whereas $J_N(p)$ is not attained for any $p\in [2,2+{\delta }_N)$. These settle some conjectures of [Gontier--Lewin--Nazar, ARMA, 2021] and [Frank--Gontier--Lewin, CMP, 2021].

arXiv commentThis is an improved version of the manuscripts arXiv:2606.16544 and arXiv:2604.17363