Explore improved Lieb-Thirring type inequalities for non-selfadjoint Schroedinger operators in this 41-minute conference talk from the Workshop on "Spectral Theory of Differential Operators in Quantum Theory" at the Erwin Schrödinger International Institute for Mathematics and Physics. Delve into the study of eigenvalues of Schroedinger operators with complex-valued potentials, examining enhancements to the inequalities established by Demuth, Hansmann, and Katriel in their 2009 Journal of Functional Analysis paper. Learn about the incorporation of a positive, integrable function in the improved result and understand why the one-dimensional case demonstrates the sharpness of this approach, highlighting the impossibility of an analogous inequality with a non-integrable function.