Explore advanced microlocal methods for operators with analytic coefficients in this comprehensive lecture delivered by Maciej Zworski at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI). Delve into the second part of a series focusing on spectral theory and mathematical relativity, presented as part of the Thematic Programme held from June 5 to July 28, 2023. Over the course of 65 minutes, gain insights into cutting-edge techniques and applications in the field of mathematical physics, building upon the foundations laid in the previous talk. Enhance your understanding of complex mathematical concepts and their relevance to modern physics research.