Differential complexes in time-periodic Gelfand-Shilov spaces

We study the global solvability of a class of differential complexes on the product manifold Tᵐ × Rⁿ T m × R n associated with systems of evolution operators of the form Lᵣ = ∂ tᵣ + iaᵣ(t)P(x,Dₓ), r=1,… ,m, L r = ∂ t r + i a r ( t ) P ( x , D x ) , r = 1 , … , m , where the coefficients aᵣ a r are real-valued Gevrey functions on the torus and P(x,Dₓ) P ( x , D x ) is a globally elliptic normal differential operator on Rⁿ 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.