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The change in mass of a nucleon, arising from its interactions with othernucleons inside the target, results in velocity-dependent terms in theSchr\"odinger equation that describes nucleon scattering. It has recently beensuggested in a number of publications that introducing and fittingvelocity-dependent terms improves the quality of the description of nucleonscattering data for various nuclei. The present paper discussesvelocity-dependent optical potentials in a context of a three-body problem usedto account for deuteron breakup in the entrance channel of $(d,p)$ reactions.Such potentials form a particular class of nonlocal optical potentials whichare a popular object of modern studies. It is shown here that because of aparticular structure of the velocity-dependent terms the three-body problem canbe formulated in two different ways. Solving this problem within an adiabaticapproximation results in a significant difference between the two approachescaused by contributions from the high $n$-$p$ momenta in deuteron in one ofthem. Solving the three-body problem beyond the adiabatic approximation mayremove such contributions, which is indirectly confirmed by replacing theadiabatic approximation by the folding Watanabe model where such contributionsare suppressed. Discussion of numerical results is carried out for the$^{40}$Ca($d,p)^{41}$Ca reaction where experimental data both on elasticscattering in entrance and exit channels and on nucleon transfer are available.

Three-body problem with velocity-dependent optical potentials: a case of $(d,p)$ reactions