S-wave resonances in positron scattering by He+

$S$-wave $e^{+}+{\mathrm{He}}^{+}$ scattering is calculated using the hyperspherical close-coupling method. The two resonances found at about $-$0.73 Ry and $-$0.39 Ry by Bhatia and Drachman [Phys. Rev. A 42, 5117 (1990)] and reproduced by Ho [Phys. Rev. A 53, 3165 (1996)] do not occur in the present results. An argument against these proposed resonances is presented on the basis of the adiabatic hyperspherical potentials. Instead of these resonances, we have found narrow Feshbach resonances just below the thresholds $(-$0.25 a.u. and $-$0.0625 a.u.) of the formation of $\mathrm{Ps}\mathrm{}(n=1\mathrm{})\mathrm{}$ and $\mathrm{Ps}\mathrm{}(n=2\mathrm{})\mathrm{}$ and an extremely narrow shape resonance just above the threshold $-$0.08 a.u. of He${}^{+}$ $(n$=5). They would be hardly detected since (i) the resonance widths are of the order of ${10}^{-4}$–${10}^{-5}$ a.u. or less, (ii) the resonance is weak in the elastic cross section, (iii) the magnitudes of the inelastic and Ps-formation cross sections are small, and (iv) the smooth cross sections due to higher partial waves should still be added to the $S$-wave cross section.