Differential Complexes in Time-Periodic Gelfand-Shilov Spaces

Abstract

We study the global solvability of a class of differential complexes on the product manifold ${\mathbb{T}}^m\times {\mathbb{R}}^n$ associated with systems of evolution operators of the form $L_r={\partial }_{t_r}+i{a}_r(t)P(x,D_x),r=1,\dots ,m,$ where the coefficients $a_r$ are real-valued Gevrey functions on the torus and $P(x,D_x)$ is a globally elliptic normal differential operator on ${\mathbb{R}}^n$. Within the framework of time-periodic Gelfand--Shilov spaces, we introduce a natural differential complex generated by these operators and investigate its solvability in both functional and ultradistributional settings. We provide a complete characterization of global solvability in terms of a Diophantine condition involving the constant part of the associated $1$-form and the spectrum of $P$. We also analyze global hypoellipticity of the complex. These results extend previous works on scalar operators and constant coefficient systems to the setting of differential complexes with time-dependent real coefficients.

16 pages