High-energy tail of the linear momentum distribution in the ground state of hydrogen atoms or hydrogen-like ions

A long-standing dispute concerning the high-energy tail of the linear momen tum distribution (HTMD) in the ground state of hydrogen atoms/hydrogen-like ions (GSHA) has been unresolved up to now.,A possible resolution of the ab ove dispute might be connected to the problem of the role of singular solut ions of quantal equations, which is a fundamental problem in its own right. The paradigm is that, even allowing for finite nuclear sizes, singular sol utions of the Dirac equation for the Coulomb problem should be rejected for nuclear charges Z < 1/alpha approximate to 137. In this paper we break thi s paradigm. First, we derive a general condition for matching a regular int erior solution with a singular exterior solution of the Dirac equation for arbitrary interior and exterior potentials. Then we find explicit forms of several classes of potentials that allow such a match. Finally, we show tha t, as an outcome, the HTMD far the GSHA acquires terms falling off much slo wer than the 1/p(6)-law prescribed by the previously adopted quantal result . Our results open up a unique way to test intimate details of the nuclear structure by performing atomic (rather than nuclear) experiments and calcul ations.