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Relation between widths of proton resonances and neutron asymptotic normalization coefficients in mirror states of light nuclei in a microscopic cluster model

It has been suggested recently ({\it Phys. Rev. Lett.} 91, 232501 (2003)) that the widths of narrow proton resonances are related to neutron Asymptotic Normalization Coefficients (ANCs) of their bound mirror analogs because of charge symmetry of nucleon-nucleon interactions. This relation is approximated by a simple analytical formula which involves proton resonance energies, neutron separation energies, charges of residual nuclei and the range of their strong interaction with the last nucleon. In the present paper, we perform microscopic-cluster model calculations for the ratio of proton widths to neutron ANCs squared in mirror states for several light nuclei. We compare them to predictions of the analytical formula and to estimates made within a single-particle potential model. A knowledge of this ratio can be used to predict unknown proton widths for very narrow low-lying resonances in the neutron-deficient region of the $sd$- and $pf$-shells, which is important for understanding the nucleosynthesis in the $rp$-process.