A non-degeneracy theorem for interacting fermions in one dimension
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arXiv
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2025
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| _version_ | 1866913021847142400 |
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| author | Corso, Thiago Carvalho |
| author_facet | Corso, Thiago Carvalho |
| contents | 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 $H_N(v,w) = -Δ+ \sum_{i\neq j}^N w(x_i,x_j) + \sum_{j=1}^N v(x_i)$ 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 $h(v) = -Δ+v$. In addition, we prove strict inequalities between the lowest eigenvalues of different self-adjoint realizations of $H_N(v,w)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_18440 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A non-degeneracy theorem for interacting fermions in one dimension Corso, Thiago Carvalho Spectral Theory Mathematical Physics Quantum Physics Primary: 81Q10 Secondary: 35J10, 81V74 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 $H_N(v,w) = -Δ+ \sum_{i\neq j}^N w(x_i,x_j) + \sum_{j=1}^N v(x_i)$ 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 $h(v) = -Δ+v$. In addition, we prove strict inequalities between the lowest eigenvalues of different self-adjoint realizations of $H_N(v,w)$. |
| title | A non-degeneracy theorem for interacting fermions in one dimension |
| topic | Spectral Theory Mathematical Physics Quantum Physics Primary: 81Q10 Secondary: 35J10, 81V74 |
| url | https://arxiv.org/abs/2503.18440 |