Structural Asymmetry and Rigidity of Bounded Solutions for a Parametrized Complex Linear Differential Equation
Walid Oukil
TL;DR
The paper analyzes a parametrized non-homogeneous complex linear ODE on $[1,\infty)$ with a complex parameter $w$ in the strip $0<\Re(w)<1$, and a real bounded forcing term $\eta$ with rotation number $\rho$. It establishes a rigidity phenomenon: for $\Re(w)\neq \tfrac{1}{2}$, the two bounded solutions corresponding to $w$ and $1-\bar w$ (both started from the same initial value) cannot both remain bounded, connecting to a Dynamical Conjecture and revealing asymmetry in parameter dependence. A central tool is a novel integral transform $\Psi_{n,c}$ that yields a contradiction unless $\Re(w)=\tfrac{1}{2}$, enabling a precise characterization of boundedness. The authors apply the result to the $\zeta$-function by taking $\eta(t)=\{t\}$, linking boundedness to Abel-sum relations and showing that certain $\zeta$-pairs cannot vanish simultaneously when $\Re(s)\neq \tfrac{1}{2}$. Overall, the work combines a rigidity analysis for complex ODEs with an intriguing bridge to analytic number theory.
Abstract
We introduce a family of parametrized non-homogeneous linear complex differential equations on $[1,\infty)$, depending on a complex parameter $w$ in the strip $\Re(w)\in(0,1)$. Their dynamics encodes subtle asymptotic features of a real bounded forcing term. Under a mild integrability condition, we establish a rigidity phenomenon: for every parameter $w$ with $\Re(w) \in (0,1)\setminus\{\tfrac12\}$, the two solutions corresponding respectively to $w$ and $1-\bar w$, both taken with initial condition $1$, cannot be simultaneously bounded on $[1,+\infty)$. This result provides an answer to the Dynamical Conjecture formulated in [arXiv:2112.05521] and highlights a structural asymmetry in the solutions' dependence on the complex parameter, thereby contributing to a better understanding of the qualitative behavior of this class of differential systems.