yields an upper bound to the polaron self-energy E < -C_{PL} \alpha^2 , valid for all α, where C_{PL} is a constant determined by solving an integro-differential equation. It was an open question for many years whether this expression was asymptotically exact as α tends to infinity. Finally, Donsker and Varadhan, applying large deviation theory to Feynman's path integral formulation for the self-energy, showed the large α exactitude of this Landau–Pekar formula. Later, Lieb and Thomas
