The integration of operator kernels with the Wigner distribution, first conceptualized by E. Wigner in 1932 and later extended by L. Cohen and others, has opened new avenues in time-frequency analysis and operator calculus.
Despite substantial advancements, the presence of "ghost frequencies" in Wigner kernels continues to pose significant challenges, particularly in the analysis of Fourier integral operators (FIOs) and their applications to partial differential
equations (PDEs).
In our study, we build on the foundational concepts of Wigner analysis to introduce a novel framework for controlling ghost frequencies through combined Gaussian and Sobolev regularization techniques. By focusing on FIOs with nonquadratic
phase functions, we develop rigorous estimates for the Wigner kernels that are crucial for their applicability to Schrödinger equations with non-trivial symbol classes. Unlike previous approaches, our methodology mitigates the interference caused by ghost frequencies and establishes robust bounds in the context of generalized symplectic mappings.