Writing the three-dimensional Laplacian in spherical coordinates and substituting psi = u(r)/r is the standard route to the reduced radial Schroedinger equation, but the usual identity for the radial part is invalid as an operator identity at the origin: it misses a singular term. Treated in the sense of distributions, the correct relation carries an extra -4*pi*delta^3(r)*u(0) contribution, exactly the term that makes the Laplacian of 1/r a point source. The consequence is that u(0)=0 is not an extra physical assumption or a normalizability argument but a necessary and sufficient condition for the reduced radial equation to be equivalent to the original three-dimensional problem - a geometric requirement imposed by the coordinate system rather than a dynamical one. For regular potentials this simply reproduces the textbook Dirichlet condition, so results derived with the reduced equation remain valid and no self-adjoint-extension machinery is needed. The correction matters for singular attractive potentials, where solutions violating u(0)=0 exist and additional bound states appear; there a one-parameter extension family (parametrized by the ratio of the additional to the standard scattering length) supplies at most one extra level, with the extra state selected by a quantum-defect criterion and, in the Coulomb Klein-Gordon case, corresponding to the disputed deeply bound levels.
Reducing the three-dimensional Schrodinger equation to a radial equation in spherical coordinates silently discards a Dirac delta term. The identity used in textbooks, (d^2/dr^2 + (2/r) d/dr) f = (1/…