Laser resonance transitions between normally metastable slates of antiprotonic helium atoms were induced making use of state dependent quenching effects caused by trace admixtures of H-2 to the target helium gas. With this method of "H-2-assisted inverse resonances" the decay rates of the stales (n, l)= (39,l), l = 36,37,38, and (38,l), l = 35,36,37 of (p) over bar He+ were determined as a function of the H-2 admixture. The quenching cross sections at 30 K deduced therefrom for the states with n = 39 were found to be of the order of the geometrical cross section for (p) over bar He+-H-2 collisons (2 10-(15) cm(2)), with a moderate decrease with increasing I. Within a given cascade with constant v = n - l - 1, the quenching cross sections for states with n = 38 are smaller by a factor of 4-6 than those for states with n = 39.