Operator Formalism for Laser-Plasma Wakefield Acceleration

In this paper, we develop an operator-based framework for laser--plasma wakefield acceleration (LPWA) in capillary discharges, providing a compact and systematic description of the coupled dynamics of laser fields and plasma response. The formalism employs key operators: the transverse modal operator hat{K}, the nonlinear plasma operator hat{N}[Ψ], the plasma oscillation operator hatΩ_p ², and the ponderomotive source operator hatα, which together describe mode coupling, plasma oscillations, and nonlinear feedback induced by the ponderomotive force. In the linear regime, the system is characterized by invariant subspaces associated with stable modal structures, while nonlinear interactions break these invariances, leading to mode mixing and complex dynamics. The approach establishes a direct connection between LPWA and Hilbert-space operator theory, including the invariant subspace, providing a formal mathematical interpretation of energy transfer and wakefield formation. Furthermore, the operator formalism integrates with neural operator methods, allowing efficient approximation of hat{N} and hatα for reduced-order modeling and predictive control. This hybrid physics--AI framework offers a robust foundation for modeling, analysis, and optimization of high-intensity laser--plasma interactions in next-generation accelerator experiments.