The potential looks much nicer, when you plot the inverses of the primes. You get one more oscillation for every new prime. So one can anticipate an asymptotically oscillating potential that encodes all the information about primes. Pretty interesting.
V = dressingTransformationPotentials[1/# & /@ Prime[Range[50]], 400];
Plot[V[Abs[x]], {x, -400, 400}]
