Apolgies in advance if this is a confusing ramble and multitude of questions, I'm not quite sure how to articulate myself. I am currently reading up on quantum mechanics and seem to have confused myself when trying to understand the difference between a discrete spectrum and continuous spectrum.
I understand the basic starting example- points on a line lying at integer values with spacing $\epsilon$ vs when $\epsilon \to 0$. I can also see how the mathematics involved differs- summation vs integration (limit of this summation).
Where I believe my confusion begins is when dealing with real examples. Am I correct in assuming that there are no conditions on the possible values of position (eigenvalues can take any values) which is why when normalising a wave-function dependent on time and position, we integrate over all of space?
However angular momentum, spin, energy- are all quantised, so are only able to take discrete values. Does the time-independent Schrodinger equation only deal with discrete values for energy? Is this the difference between continuous and discrete- simply whether there are constraints placed on the eigenvalues and hence the wave function?
And how does this relate to the potential energy and its implications of bounded vs scattering states? Are bounded states analogous to discreteness?
If anybody could clear up any of the above I'd greatly appreciate it. I thought I had some grasp on the topic until I started thinking harder.