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Adiabatic model of $(d,p)$ reactions with explicitly energy-dependent non-local potentials

We have developed an approximate way of dealing with explicit energy-dependence of non-local nucleon optical potentials as used to predict the $(d,p)$ cross sections within the adiabatic theory. Within this approximation, the non-local optical potentials have to be evaluated at an energy shifted from half the incident deuteron energy by the $n-p$ kinetic energy averaged over the range of the $n-p$ interaction and then treated as an energy-independent non-local potential. Thus the evaluation of the distorting potential in the incident channel is reduced to a problem solved in our previous work in [{\it Phys. Rev. Lett. 110, 112501(2013) and Phys. Rev. C 87, 064610 (2013)}]. We have demonstrated how our new model works for the case of $^{16}$O$(d,p)^{17}$O, $^{36}$Ar($d,p)^{37}$Ar and $^{40}$Ca$(d,p)^{41}$Ca reactions and highlighted the need for a detailed understanding of energy-dependence of non-local potentials. We have also suggested a simple way of correcting the $d-A$ effective potentials for non-locality when the underlying energy-dependent non-local nucleon potentials are unknown but energy-dependent local phenomenological nucleon potentials are available.