In 1D quantum mechanics problems, the energy spectrum is often determined by the limits of the potential at $\pm$ infinity. Generally, the spectrum is continuous non-degenerate when energy is above both limits, continuous and degenerate when in between, and discrete and degenerate when below both. Often we can even see the transition between the discrete and continuous regimes when we look at, for example, the hydrogen atom. Below $0$, the spectrum is discrete but the levels get infinitely close as you approach positive values and then continuous for the positive values. This is because the potential energy limit on both sides is $0$.
Continuing this investigation, I wanted to know what would happen in potentials without a well defined limit like for example a sinusoidal potential? It seems like a fairly difficult question because there is a distinct qualitative difference between taking the top of the sine as your limit and the bottom so a process of finding the limit of the energy spectrum as you extend the sine wave to infinity seems like it would surely fail. Please let me know.