  We analyze how a short distance boundary condition for the Schrodinger
equation must change as a function of the boundary radius by imposing the
physical requirement of phase shift independence on the boundary condition. The
resulting equation can be interpreted as a variable phase equation of a
complementary boundary value problem. We discuss the corresponding infrared
fixed points and the perturbative expansion around them generating a short
distance modified effective range theory. We also discuss ultraviolet fixed
points, limit cycles and attractors with a given fractality which take place
for singular attractive potentials at the origin. The scaling behaviour of
scattering observables can analytically be determined and is studied with some
emphasis on the low energy nucleon-nucleon interaction via singular pion
exchange potentials. The generalization to coupled channels is also studied.
