What potentials permit a uniqueness theorem
David F. Bartlett, Yu Su · American Journal of Physics · 1994
We show that the Yukawa and Coulomb potentials are the only two real, central, analytic potentials that permit a uniqueness theorem in all bounded geometries. Only for point potentials of the form U(r)=(α/r)cosh(kr)+(β/r)sinh(kr) is it possible to construct a potential V(r) everywhere within any bounded region given V at every point on the boundary (Dirichlet conditions). The replacement of the hyperbolic by trigonometric functions gives a class of ‘‘sinusoidal’’ potentials. These generally have an associated uniqueness theorem, but fail to do so when the boundary forms a resonant cavity for the particular k.