  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.
