The kaon energy in neutron matter is calculated analytically with the Klein-Gordon equation, by making a Wigner-Seitz cell approximation and employing a $K^-N$ square well potential. The transition from the low density Lenz potential, proportional to scattering length, to the high density Hartree potential is found to begin at fairly low densities. Exact non-relativistic calculations of the kaon energy in a simple cubic crystal of neutrons are used to test the Wigner-Seitz and the Ericson-Ericson approximation methods. All the calculations indicate that by $\sim 4$ times nuclear matter density the Hartree limit is reached, and as the Hartree potential is less attractive, the density for kaon condensation appears to higher than previously estimated. Effects of a hypothetical repulsive core in the $K^-N$ potential are also studied.

4 pages including 2 figures