Cubic Nonlinear Schrödinger Systems and Orthonormal Fermionic Ground States in ℝ
Abstract
We study the one-dimensional nonlinear Schrödinger system −ui''−2(∑k=1N uk2)p−1ui=μi ui in ℝ, i=1,2,⋯,N, tagS where u=(u1,⋯,uN)∈ H1(ℝ)N, μ1≤μ2≤⋯≤μN<0, 1<p<3 and N≥ 2 are arbitrary. For the cubic case where p=2 and N≥ 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 L2-mass formulas of all nontrivial solutions are derived in terms of distinct Lagrange multipliers μi. 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≥2. We then investigate a fermionic variational problem JN(p) with N≥2, whose minimizers correspond to ground states of the system (S) under orthonormal constraints. More precisely, we prove that there exists a sufficiently small δN>0 such that JN(p) is attained for any p∈(2−δN,2), whereas JN(p) is not attained for any p∈[2,2+δN). These settle some conjectures of [Gontier--Lewin--Nazar, ARMA, 2021] and [Frank--Gontier--Lewin, CMP, 2021].
Related
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BibTeX
@misc{chen2026cubic,
title = {{Cubic Nonlinear Schrödinger Systems and Orthonormal Fermionic Ground States in $\mathbb R$}},
author = {Bin Chen and Yujin Guo and Yong Luo and Juncheng Wei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17941},
eprint = {2610.17941},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17941},
}
RIS
TY - UNPB
TI - Cubic Nonlinear Schrödinger Systems and Orthonormal Fermionic Ground States in ℝ
AU - Bin Chen
AU - Yujin Guo
AU - Yong Luo
AU - Juncheng Wei
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17941
UR - https://arxiv.org/abs/2610.17941
AB - We study the one-dimensional nonlinear Schrödinger system −u_i''−2(∑_k=1^N u_k^2)^p−1u_i=μ_i u_i in ℝ, i=1,2,⋯,N, tagS where u=(u_1,⋯,u_N)∈ H^1(ℝ)^N, μ_1≤μ_2≤⋯≤μ_N<0, 1<p<3 and N≥ 2 are arbitrary. For the cubic case where p=2 and N≥ 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 μ_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≥2. We then investigate a fermionic variational problem J_N(p) with N≥2, whose minimizers correspond to ground states of the system (S) under orthonormal constraints. More precisely, we prove that there exists a sufficiently small δ_N>0 such that J_N(p) is attained for any p∈(2−δ_N,2), whereas J_N(p) is not attained for any p∈[2,2+δ_N). These settle some conjectures of [Gontier--Lewin--Nazar, ARMA, 2021] and [Frank--Gontier--Lewin, CMP, 2021].
ER -