On the momentum dependence of the nuclear potential

Abstract

It is shown that the potential energy of a nucleon in the nucleus, evaluated in the first approximation of the perturbation method or on the basis of Brueckner’s theory, obeys a hyperbolic partial differential equation, which is independent of any nuclear parameter and is established by the antisymmetry properties of the nucleon assembly only. This nuclear equation rules the dependence of the potential both on the nucleon momentum and the nuclear density. The particular choice of the two-body forces or of the nucleon-nucleon phaseshifts for the evaluation of the potential energy of the nucleus, implies a specialization of the Cauchy problem, related to this equation, regardless of the saturation prescriptions of nuclear forces. It is shown that there exists a class of solutions of this nuclear equation which cannot be derived, in the first approximation of the perturbation method, from any of the known two-body potentials. This class of solutions, however, leads to the saturation of the nuclear binding energy and density as well as to the experimental value of the symmetry energy and to the correct behavior, in the low energy region, of the real and imaginary parts of the nuclear potential. A mathematical proof is given of the dependence of the potential inside the Fermi sphere on even powers of the nucleon momentum, as required by the invariance prescription of the potential with respect to time reflection. The linear dependence of the potential on the square of the nucleon momentum, and the so called nucleon effective mass approximation, is discussed in the light of the correspondence principle, which has been used to describe the motion of a nucleon in nuclear matter. Finally, it is shown that the Johnson-Teller, Schiff-Thirring and Drell-Huang theories of nuclear saturation implicitly involve only particular solutions of the considered nuclear equation.