Explore a comprehensive lecture on the complete trans-series for energy densities in integrable models, presented by Zoltan Bajnok at the Galileo Galilei Institute. Delve into advanced concepts in mathematical physics and gain insights into the intricate relationships between energy densities and integrable systems. Over the course of 1 hour and 20 minutes, examine the theoretical foundations and practical applications of trans-series in the context of integrable models, enhancing your understanding of this complex topic in theoretical physics.