On maximal multiplicities for Hamiltonians with separable variables

For $\mathbb N^*:=\mathbb N \setminus \{0\}$, we consider the collection $\mathfrak M(N)$ of all the $N$ rows, for which, for $n=1,\cdots,N$, the $n-th$ row consists of an increasing sequence $(a_j^n)_j$ of real numbers. For $\mathfrak A \in \mathfrak M(N)$, we define its spectrum $\sigma(\mathfrak A)$ by $\sigma(\mathfrak A)=\{\lambda\in \mathbb R \;|\; \lambda=\sum_{n=1}^Na_{j_n}^n\}\,,$ where $(j_1,j_2,\dots,j_N)\in (\mathbb N^*)^N$. This spectrum is discrete and consists of an infinite sequence that can be ordered as a strictly increasing sequence $\lambda_k(\mathfrak A)$. For $\lambda \in \sigma (\mathfrak A)$ we denote by $m(\lambda,\mathfrak A)$ the number of representations of such a $\lambda$, hence the multiplicity of $\lambda$.\\ In this paper we investigate for given $N\in \mathbb N^*$ and $k\in \mathbb N^*$ the highest possible multiplicity (denoted by $\mathfrak m_k(N)$) of $\lambda_k(\mathfrak A)$ for $\mathfrak A \in \mathfrak M(N)$. We give the exact result for $N=2$ and for $N=3$ prove a lower bound which appears, according to numerical experiments, as a "good" conjecture. For the general case, we give examples demonstrating that the problem is quite difficult. \\ This problem is equivalent to the analogue eigenvalue multiplicity questions for Schr\"odinger operators describing a system of N non-interacting one-dimensional particles.