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A non-degeneracy theorem for interacting electrons in one dimension

Thiago Carvalho Corso

TL;DR

This work establishes that the ground-state of one-dimensional interacting electron systems with distributional external and interaction potentials is non-degenerate and non-vanishing almost everywhere under Fermi statistics. The authors develop a Perron–Frobenius theorem for distributional Schrödinger operators and a novel unitary reduction to the simplex, which together bypass the obstacles posed by antisymmetry. They prove non-degeneracy for local boundary conditions and, under a parity condition, for non-local boundary conditions, and derive strong unique continuation for the single-particle operator, along with eigenvalue inequalities for different self-adjoint realizations. The results provide a rigorous foundation for 1D electronic models and have immediate implications for sharp spectral inequalities and, potentially, v-representability in 1D density functional theory.

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 $H_N(v,w) = -Δ+ \sum_{i\neq j}^N w(x_i,x_j) + \sum_{j=1}^N v(x_i)$ 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 $H_N(v,w)$.

A non-degeneracy theorem for interacting electrons in one dimension

TL;DR

This work establishes that the ground-state of one-dimensional interacting electron systems with distributional external and interaction potentials is non-degenerate and non-vanishing almost everywhere under Fermi statistics. The authors develop a Perron–Frobenius theorem for distributional Schrödinger operators and a novel unitary reduction to the simplex, which together bypass the obstacles posed by antisymmetry. They prove non-degeneracy for local boundary conditions and, under a parity condition, for non-local boundary conditions, and derive strong unique continuation for the single-particle operator, along with eigenvalue inequalities for different self-adjoint realizations. The results provide a rigorous foundation for 1D electronic models and have immediate implications for sharp spectral inequalities and, potentially, v-representability in 1D density functional theory.

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 acting on , where the external and interaction potentials and 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 and . 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 . In addition, we prove strict inequalities between the lowest eigenvalues of different self-adjoint realizations of .

Paper Structure

This paper contains 25 sections, 26 theorems, 131 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Notation
  4. Main results
  5. Outline of the proofs
  6. Background on Sobolev Spaces
  7. Definitions and basic properties
  8. Representation and density on dual Sobolev spaces
  9. Regularity of reduced densities
  10. Schrödinger operators with distributional potentials
  11. Self-adjoint realizations
  12. From single-particle to many-particle boundary conditions
  13. Examples of distributional potentials
  14. Non-degeneracy of the ground-state
  15. Perron-Frobenius theorem
  16. ...and 10 more sections

Key Result

Theorem 2.1

Let $v\in \mathcal{V}$, $w\in \mathcal{W}$, $N\in \mathbb N$ and $\Gamma \subset \partial \Omega$. Then the operator $H_{N}(v,w;\Gamma)$ defined as the self-adjoint realization of eq:quadratic form0 with form domain $\mathrm{H}^1(I_N;\Gamma) \cap \mathcal{H}_N$ has a unique ground-state $\Psi$ and $

Figures (2)

  • Figure 1: Simplex $S_N$ in the case $N=2$ (left) and $N=3$ (right) with edges of $\partial I_N$ (dashed lines) and interior boundary $\Gamma_{\rm int} \subset \partial S_N$ (in red).
  • Figure 2: Example of extended set $\Omega$ in the case $N=2$ (left) and $N=3$ (right) with interior boundary $\Gamma_{\rm int}$ (in red), exterior boundary $\partial \Omega \setminus \Gamma_{\rm int}$ (in blue), and $B\cap \partial S_N \subset F_N$ (in yellow).

Theorems & Definitions (70)

  • Theorem 2.1: Non-degeneracy theorem with local BCs
  • Remark 2.2: Unique continuation
  • Theorem 2.3: Non-degeneracy theorem with non-local BCs
  • Remark 2.4: On the condition on the number of particles
  • Remark 2.5: Further extensions
  • Theorem 2.6: Single-particle operators
  • Remark : Self-adjont realizations
  • Remark : Sturm-Liouville theory
  • Theorem 2.7: Monotonicity with respect to the Dirichlet set
  • Remark : Strict inequality
  • ...and 60 more