Quenched post-starburst galaxies (QPSBs) are a rare but important class of galaxies that show signs of rapid cessation or recent rejuvenation of star formation. A recent observation shows that about half of QPSBs have large amounts of cold gas. This molecular CO sample is, however, too small and is not without limitations. Our work aims to verify previous results by applying a new method to study a uniformly selected sample, more than 10 times larger. In particular, we present detailed analysis of H$alpha$/H$beta$ ratios of face-on QPSBs at $z = 0.02 - 0.15$ and with $M_star = 10^{10}-10^{11},M_odot$. We interpret the H$alpha$/H$beta$ ratios by applying our recent gas mass calibration, which is based on non-PSB galaxies but predicts gas masses that are consistent with CO observations of $sim 100$ PSBs. We estimate the molecular gas by either using PSBs with well-measured H$alpha$/H$beta$ ratios or by measuring them from stacked spectra. Our analysis reveals that QPSBs have a wide range of H$alpha$/H$beta$ ratios and molecular gas fractions that overlap with the typical gas fractions of star-forming or quiescent galaxies: H$alpha$/H$beta approx 3-8$ and $f_mathrm{H_2} approx 1%-20%$ with median $f_mathrm{H_2} approx 4%-6%$, which correspond to $M_mathrm{H_2} approx (1-3) times 10^{9} ,M_odot$. Our results indicate that large reservoirs of cold gas are still present in significant numbers of QPSBs, and that they arguably were not removed or destroyed by feedback from active galactic nuclei.
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