Abstract
A long-standing dispute concerning the high-energy tail of the linear momentum distribution (HTMD) in the ground state of hydrogen atoms/hydrogen-like ions (GSHA) has been unresolved up to now. A possible resolution of the above dispute might be connected to the problem of the role of singular solutions of quantal equations, which is a fundamental problem in its own right. The paradigm is that, even allowing for finite nuclear sizes, singular solutions of the Dirac equation for the Coulomb problem should be rejected for nuclear charges Z < 1/α≈137. In this paper we break this paradigm. First, we derive a general condition for matching a regular interior 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 that, as an outcome, the HTMD for the GSHA acquires terms falling off much slower than the 1/p6-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 calculations.