A non-degeneracy theorem for interacting fermions in one dimension

Abstract

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 $H_N(v,w)=-\Delta +{\sum }_{i\ne j}^{N}w(x_i,x_j)+{\sum }_{j=1}^{N}v(x_i)$ acting on ${\wedge }^N{\mathrm{L}}^2([0,1])$, 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 $H_N(v,w)$.

Revised version to appear in Annales Henri Poincaré

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