  We study the quantum reflection of ultracold antihydrogen atoms bouncing on
the surface of a liquid helium film. The Casimir-Polder potential and quantum
reflection are calculated for different thicknesses of the film supported by
different substrates. Antihydrogen can be protected from anni- hilation for as
long as 1.3s on a bulk of liquid 4He, and 1.7s for liquid 3He. These large
lifetimes open interesting perspectives for spectroscopic measurements of the
free fall acceleration of antihydrogen. Variation of the scattering length with
the thickness of a film of helium shows interferences which we interpret
through a Liouville transformation of the quantum reflection problem.
