Sometimes, there is an absolute limit on the number of terms an officeholder can serve, while, in other cases, the restrictions are merely on the number of consecutive terms.
For example, Maxwell's Equations would always be capitalized, while quantum mechanics is usually un-capitalized. I searched half a dozen sites for an answer to this but found no clue about a rule. So go either way that makes you happy, but (most of all) be consistent.
In a real structure (which is not perfectly 2-D), there are finite energy ranges over which the energy independence holds (the derivation holds for each single, well separated possible value of k z). The resulting density of states for a quantum well is a staircase, as below in red.