On maximal multiplicities for Hamiltonians with separable variables

Abstract

For ${\mathbb{N}}^{\ast }:=\mathbb{N}\setminus \{0\}$, 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 $\sigma (\mathfrak{A})=\{\lambda \in \mathbb{R}|\lambda ={\sum }_{n=1}^N{a}_{j_n}^n\},$ where $(j_1,j_2,\dots ,j_N)\in ({\mathbb{N}}^{\ast }{)}^N$. 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 $N\in {\mathbb{N}}^{\ast }$ and $k\in {\mathbb{N}}^{\ast }$ 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 Schrödinger operators describing a system of N non-interacting one-dimensional particles.