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On gap properties for the linearized 1D Dirac--Soler model

Danko Aldunate, Julien Ricaud, Edgardo Stockmeyer

Abstract

We study spectral properties of the Dirac operator $L_0$ arising as the upper-right off-diagonal block in the linearization around standing wave solutions of the one-dimensional Soler model with power nonlinearity $f(s)=s|s|^{p-1}$, $p>0$. Our main results concern the so-called gap property: we show that if $p \geq 1$, then the only eigenvalues of $L_0$ are its ground state energies, $-2ω$ and $0$. In contrast, for $p<1$, additional eigenvalues appear from the thresholds of the essential spectrum. Furthermore, we prove that the thresholds never admit eigenvalues and that they have at most one resonance.

On gap properties for the linearized 1D Dirac--Soler model

Abstract

We study spectral properties of the Dirac operator arising as the upper-right off-diagonal block in the linearization around standing wave solutions of the one-dimensional Soler model with power nonlinearity , . Our main results concern the so-called gap property: we show that if , then the only eigenvalues of are its ground state energies, and . In contrast, for , additional eigenvalues appear from the thresholds of the essential spectrum. Furthermore, we prove that the thresholds never admit eigenvalues and that they have at most one resonance.

Paper Structure

This paper contains 9 sections, 21 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Strategy of the proofs & other important results
  4. Theorem \ref{['main_thm_1_gap']}: the min-max principle
  5. Theorem \ref{['main_thm_2_noEV']} & Theorem \ref{['main_thm_3_Resonances']}: characterizing generalized eigenvalues at thresholds
  6. Propagation in p of the gap property: proof of Proposition \ref{['main_propagation_prop_global']}
  7. Existence of an additional eigenvalue in the gap when p<1: proof of Proposition \ref{['Presence_of_other_ev_for_p_less_than_1']}
  8. Absence of eigenvalues at the thresholds: proofs of Proposition \ref{['Proposition_Property_resonances_at_thresholds']} & Theorem \ref{['main_thm_2_noEV']}
  9. Simplicity of resonances at the thresholds: proofs of Theorems \ref{['Thm_at_most_one_generalized_eigenfunction_at_thresholds']} & \ref{['main_thm_3_Resonances']}

Key Result

Theorem 1

Let $(p,\omega)\in(0,+\infty)\times(0,m)$ and $L_0$ be as in Def_L_mu with $f(s)=s|s|^{p-1}$.

Theorems & Definitions (49)

  • Theorem 1: Characterization (in $p$) of the absence of eigenvalues strictly inside the spectral gap
  • Remark
  • Theorem 2: Absence of eigenvalues at the thresholds
  • Theorem 3: Simplicity of resonances at the thresholds
  • Corollary 4
  • Proposition 5
  • Proposition 6
  • Proposition 7: Properties of generalized eigenfunctions at the thresholds
  • Remark
  • Theorem 8: Simplicity of generalized eigenvalues at the thresholds
  • ...and 39 more