What Is the Cardinality of Bloch Waves in Quantum Mechanics?

[jostpuur]

According to the Bloch's theorem, the solutions of SE in a periodic potential may be written as superpositions of Bloch waves. But what kind of superpositions are these? There is the continuous wave vector parameter, over which we can integrate just like in forming free wave packets, but what about the cardinality of solutions for some fixed wave vector. Are the wave equations for fixed wave vector always like this

$$\psi(x) = \sum_j u_j(x)e^{ikx},$$

where $u_j(x)$ is some sequence (or finite amount) of periodic functions, or could they be like this

$$\psi(x) = \int d\alpha\; u(x,\alpha) e^{ikx}$$

where we have periodic functions $x\mapsto u(x,\alpha)$ for each $\alpha$, where $\alpha$ is a continuous parameter?

EDIT: Now I see that this is a very strange question, because those superpositions of periodic functions (with same periodicity) are still periodic. I'm not sure what I was thinking... I think I was thinking about how to write superpositions in respect to the wave vector k, but then changed the question while typing this. Can there even be several periodic parts u(x) for some fixed wave vector? Perhaps I should start writing these questions on paper before posting them?

 

[jostpuur]

The wave vector k is not directly related to the energy with Bloch waves. How many different Bloch waves

$$\psi_j(x) = u_j(x) e^{ik_j x}$$

can there be corresponding to the same energy? Could there be eigenstates of the Hamiltonian which have the form

$$\psi(x) = \int d\alpha\; \psi_{\alpha}(x) = \int d\alpha\; u(x,\alpha) e^{ik(\alpha) x}?$$