Spectral properties in Solid state physics
So assume we have a periodic 1d <PERSON> operator $$- f'' + V(x) f(x)= \lambda f(x)$$ and we want $V$ to be periodic.
Now if we assume that we are on a finite interval and that we have periodic boundary conditions where $R$ denotes the period of the potential, then we have eigenvalues $E_0 < E_1 \le E_2 < E_3 \le E_4...$ and so on.
Okay, this is clear to me. Then, there is the case that such an operator is defined on the full interval. First question: Do we then need any boundary conditions? In my physics lecture we used so-called Born von Karmann boundary conditions (saying that $f(x+R) =f(x)$) in order to "prove" the <PERSON> or <PERSON> theorem which says that we can decompose $f(x) = e^{ikx} u_k(x)$. 346
This theorem says that we can decompose the wavefunction in a periodic part$ u_k(x+R) = u_k$ and a complex exponential $e^{ikx}$.
I somehow feel as if these Born von Karmann boundary conditions are not necessary in the sense that any eigenfunction to this <PERSON> operator is automatically periodic with the potential's period, is this true?- In that case: Why do we want <PERSON> boundary conditions?- My problem with the Born von Karmann conditions is that I find that they are not really boundary conditions, as they don't act on some boundary. So what about the domain of such an operator?
2.) In my physics lecture we noticed that due to these Born von Karmann conditions the possible $k'$s for the problem (appearing in the exponentials) are discrete. Not sure if this is automatically satisfied, even if we don't assume Born von Karmann boundary conditions? Then we said that for every $k$, the <PERSON> operator equation that you get by pluggin in the ansatz from the Bloch or Floquet thoerem has a discrete spectrum. Is this true? If so, what does this all have to do with bands, if everything is so nicely discrete? - Or do we just cal these things bands, since the $k$'s get so close, that we cannot really resolve the discrete structure?
3.) Is there any relationship between the finite-interval problem and the infinite interval problem or are these two completely different things? 346