Proof that the hydrogen-antihydrogen molecule is unstable.

In the framework of nonrelativistic quantum mechanics we derive a necessary condition for four Coulomb charges (${m}_{1}^{+},{m}_{2}^{\ensuremath{-}},{m}_{3}^{+},{m}_{4}^{\ensuremath{-}}$), where all masses are assumed finite, to form the stable system. The obtained stability condition is physical and is expressed through the required minimal ratio of Jacobi masses. In particular, this provides the rigorous proof that hydrogen-antihydrogen and muonium-antimuonium molecules and hydrogen-positron-muon systems are unstable. It also proves that replacing hydrogen in the hydrogen-antihydrogen molecule with its heavier isotopes does not make the molecule stable. These are the first rigorous results on the instability of these systems.