Let's say we have a piece-wise differentiable periodic potential. For example, like a zigzag. Let's assume that we know eigenvalues and eigenfunctions on each interval, $\Psi_{1, 2}$ on the left and $\Psi_{3, 4}$ on the right. How do we find the general solution?
For a periodic array of finite potential wells I would say that the wave function and its derivative must be continuous at the boundaries of each pit, but the second derivative can have discontinuity because the potential has a discontinuity. However, it is not the case for a continuous potential. Do I also require that the second derivatives are equal? But that implies more equations than I have eigenfunctions for a given energy.
In case of a potential like in picture, the functions on the left are
$$ \Psi_{left}(x) = C_{1} \Psi_{1}(x) + C_{2} \Psi_{2}(x),\\ \Psi_{right}(x) = C_{3} \Psi_{3}(x) + C_{4} \Psi_{4}(x). $$
What I should do? Obliviously, the functions are continuous, so
$$ C_{1} \Psi_{1}(0) + C_{2} \Psi_{2}(0) = C_{3} \Psi_{3}(0) + C_{4} \Psi_{4}(0) $$
Their derivatives also must be continuous:
$$ C_{1} \Psi_{1}^{\prime}(0) + C_{2} \Psi_{2}^{\prime}(0) = C_{3} \Psi_{3}^{\prime}(0) + C_{4} \Psi_{4}^{\prime}(0) $$
But what is my next step? Do I also require second derivatives to be continuous,
$$ C_{1} \Psi_{1}^{\prime\prime}(0) + C_{2} \Psi_{2}^{\prime\prime}(0) = C_{3} \Psi_{3}^{\prime\prime}(0) + C_{4} \Psi_{4}^{\prime\prime}(0) ? $$
But in this case, when I apply Bloch theorem, do I also apply constraints on the function derivatives?
$$ \left[C_{1} \Psi_{1}(-L/2) + C_{2} \Psi_{2}(-L/2)\right] e^{ikL} = C_{3} \Psi_{3}(L/2) + C_{4} \Psi_{4}(L/2) $$ ...and the same for the first and the second derivative? I would end up with 6 equations on 4 coefficients in this case. However, I cannot allow second derivative to have discontinuity anywhere because the potential doesn't have discontinuities. What am I doing wrong?