Why is there a discontinuity in the DOS for the lower eigenvalue here? I don’t really understand what they say in the solutions about the 4 k points below E0 belonging to E- and above E0 belonging to both E- and E+… I would say each eigenvalue has its own parabola with two k points at every given energy.
The subtle point is that E_-(k) is not one independent parabola for all k. It is the lower branch after diagonalizing the two-state problem, check also the solution to the earlier question.
Before the crossing is opened, you can think of two shifted parabolas. Below
E_0=\frac{\hbar^2 k_0^2}{2m},
a horizontal line intersects both parabolas twice, and all four states belong to the lower branch E_-.
Above E_0, the two inner intersections belong to the upper branch E_+, while only the two outer intersections remain on E_-. So the number of k-states contributing to g_-(E) drops from four to two.
Since the 1D DOS counts the number of k-solutions weighted by 1/|dE/dk|, this gives a factor-of-two drop in the partial DOS g_-(E). The discontinuity is from the branch assignment changing at the crossing, not from a single parabola changing shape.