In this paper, we show that the ground-state of many-body Schrödinger operators for electrons in one dimension is non-degenerate. More precisely, we consider Schrödinger operators of the form \(\begin{aligned} H_N(v,w) = -\Delta + \sum _{i\ne j}^N w(x_i,x_j) + \sum _{j=1}^N v(x_i) \quad \text{ acting } \text{ on } \bigwedge ^N \textrm{L}^2([0,1]), \end{aligned}\) where the external and interaction potentials v and w belong to a large class of distributions. In this setting, we show that the ground-state of the system with Fermi statistics and local boundary conditions is non-degenerate and does not vanish on a set of positive measure. In the case of periodic and anti-periodic (or more general non-local) boundary conditions, we show that the same result holds whenever the number of particles is odd and even, respectively. This non-degeneracy result seems to be new even for regular potentials v and w. As an immediate application of this result, we prove eigenvalue inequalities and the strong unique continuation property for eigenfunctions of the single-particle one-dimensional operators \(h(v) = -\Delta +v\) . In addition, we prove strict inequalities between the lowest eigenvalues of different self-adjoint realizations of \(H_N(v,w)\) .
