Outside the well a neutron's radial function is u(r) = b·W(2κr), κ = √(2μS)/ħ, with W = e−κr(1 + 1/κr) for l = 1 and e−κr(1 + 3/κr + 3/κ²r²) for l = 2. As S → 0 it tends to the zero-energy solution, which outside the well falls like r−l.
l = 0: u is flat outside, its norm diverges, so the tail carries all the probability: ⟨r²⟩ → 1/(2κ²) = ħ²/(4μS). The rms radius grows like S−1/2.
l = 1: u ~ A/r is normalizable but ∫r²u² is not: b → Aκ and ⟨r²⟩ → (5/4)A²/κ, so the rms radius grows like S−1/4.
l ≥ 2: u ~ A/rl, ∫r²u² converges: the radius stays finite at threshold.
protons: the Coulomb barrier makes the zero-energy tail fall faster than any power, and every l stays finite. This is why proton halos are rare.
All four laws are checked on the live solver down to S = 10−4 MeV (slopes of ln⟨r²⟩ against ln S: −0.991, −0.498, −0.004, −0.000; the rms radius has half these, −0.50 and −0.25).