Two pints to be clarified: 1. How is $U_c$ appears in the function $U(x)$? I assumed that it is a parameter in the function $U(x)$, or an integration form of it. 2. How can you change this parameter in real world. There will relate to the nature of the process - adiabatic.

What is $U_c$ related to the exchange potential $U(x)$? and how can you change $U_c$?

Yes. Picking only positive exponent is lossing some generality. The text you linked is giving an example of travleling wave. If you wane to show a standing wave, then the negative exponent has to participate. It is nothing but a preferential choice of the author,

Can you show $H_K$ is self-adjoint? It contians a term of $ik$, appears not Hermitian.

For a parallel thinking: can you draw a 4-d volume??