arXiv2026
We formulate a theory of pseudogap formation generated by dynamic finite nesting vector ${\bf Q}$ fluctuations with characteristic oscillation frequency $ω_0$, and damping $γ$. Starting from a Hamiltonian describing electrons coupled to a classical Gaussian random field, we introduce the double - series representations of the single - particle Green's function within an Abelian (commuting) approximation to the exact SU(2) time evolution of the pseudogap problem. The resulting propagator naturally acquires a generalized Bogoliubov structure in which every stochastic scattering history is characterized by an effective dynamic gap, leading to a coherent superposition of dynamically broadened sidebands with complex Poisson weights. A central result of the theory is the emergence of a dynamically generated decoherence scale $Γ_{\rm eff}$ governing the crossover between two qualitatively different pseudogap regimes. For $ω_0>Γ_{\rm eff}$ the fluctuating field is resolved coherently and the double-series representation provides a controlled description of dynamic sideband formation. Conversely, when $Γ_{\rm eff}\gtrsimω_0$, coherence is progressively lost and the theory crosses over to the quasistatic fluctuating - gap regime described by the exact continued-fraction solution. The coherent and quasistatic descriptions are therefore interpreted as two complementary asymptotic limits of the same microscopic dynamic pseudogap model. The detailed results of numerical calculations for electron spectral density and density of states supporting our approach are presented for different sets of model parameters confirming the general picture of this crossover.