Elliott H. Lieb: The 36 Year Old Saga of the BMV Conjecture

In 1975 D. Bessis, P. Moussa and M. Villani in JMP 16, p.2318 noted that the Feynman-Kac integral representation for Trace exp(A - t B) (with A = a Schroedinger operator and B = a perturbing potential), when considered as a function of the coupling constant t, is the Laplace transform of a positive measure. This fact has several implications, one being that it leads to a sequence of upper and lower bounds to the free energy. This posivity property holds for a many-boson system but not obviously for a many-fermion system because of the "sign problem". Boldly, they conjectured that the positivity property would hold for *any* pair of self-adjoint operators, A and B, on *any* Hilbert space. For 36 years this conjecture was attacked by many people but no proof or counter-example was found, not even for 3x3 matrices! Finally, in 2011 H. Stahl announced a proof (arXiv 1107.4875). Some of the implications of the Stahl theorem will be discussed. It is hoped that more useful implications can be found by the listeners. (Joint work with R. Seiringer).