Non-Self-Adjoint Spectral Asymptotics

The spectral theory of non-self-adjoint differential operators constitutes one of the most challenging and rapidly evolving frontiers in modern mathematical analysis. Unlike their self-adjoint counterparts, which enjoy the stability guaranteed by the spectral theorem and the reality of eigenvalues, non-self-adjoint operators exhibit highly volatile spectral behaviors, characterized by the phenomenon of spectral instability and the divergence of the pseudospectrum from the spectrum. This manuscript provides a rigorous, comprehensive examination of the spectral asymptotics of non-self-adjoint semiclassical pseudodifferential operators, with a specific focus on the distribution of eigenvalues in the semiclassical limit h→0h \to 0h→0.