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Rigidity and Structural Asymmetry of Bounded Solutions

Walid Oukil

Abstract

In this manuscript, we introduce a family of parametrized non-homogeneous linear complex differential equations on $[1,\infty)$, depending on a complex parameter. We identify a "Rotation number hypothesis" on the non-homogeneous term, which establishes a structural asymmetry: if two solutions with the same initial condition equal to $1$, corresponding respectively to the parameters $s$ and $1-\overline{s}$ lying in the critical strip, are both bounded on $[1,+\infty)$, then $\Re(s) = \tfrac{1}{2}$.

Rigidity and Structural Asymmetry of Bounded Solutions

Abstract

In this manuscript, we introduce a family of parametrized non-homogeneous linear complex differential equations on , depending on a complex parameter. We identify a "Rotation number hypothesis" on the non-homogeneous term, which establishes a structural asymmetry: if two solutions with the same initial condition equal to , corresponding respectively to the parameters and lying in the critical strip, are both bounded on , then .
Paper Structure (3 sections, 4 theorems, 63 equations)

This paper contains 3 sections, 4 theorems, 63 equations.

Table of Contents

  1. Main result
  2. Main proposition
  3. Proof of the Theorem \ref{['MainTheo']}

Key Result

Lemma 3

Let $\eta \in L^\infty(\mathbb{C})$ and $w \in \mathbb{C}_+$. Consider the continuous solution $\psi_{\eta,w}(\cdot)$ of the differential equation EDO, as introduced in Notation NotationPsi. Then, where the function $\mu_\eta$ is defined in Notation Notationmu. Furthermore, if $\eta$ satisfies the rotation number hypothesis NombreRotation, then where $\rho$ denotes the rotation number of $\eta$.

Theorems & Definitions (8)

  • Lemma 3
  • proof
  • Theorem 4
  • Proposition 6
  • proof
  • Lemma 7
  • proof
  • proof : Proof of the Theorem \ref{['MainTheo']}