Suppose $p, q : \mathbb{R} \to \mathbb{R}$ are piecewise continuous and periodic of period $a > 0$. Moreover, suppose $p$ is nowhere zero and $p'$ is piecewise continuous.
I would like to understand the spectrum of the operator $$Pf(x) = (p(x) f'(x))' + q(x) f(x),$$ acting on a suitable Hilbert space of functions, chosen so that the spectrum of $P$ consists precisely of countably many eigenvalues.
For instance, the following vague description, which I would like to make more precise, is given in Eastham's little book The spectral theory of periodic differential equations (the equation (2.1.1) that he refers to is precisely the operator $P$ I have written above).
I would like to answer the following:
What exactly is the Hilbert space $H$ on which we take $P$ to act? Eastham seems to suggest that we consider continuous functions equipped with $L^2$ inner product, but surely this is not a closed space?
What is the domain $D(P)$ of the operator $P$? Eastham seems to suggest that the domain are those functions $f$ (extended periodically to all of $\mathbb{R}$) that belong to $C^2[0,a]$ and obey $f(0)= f(a)$, $f'(0) = f'(a)$.
Once we have settled on an underlying Hilbert space $H$ and domain $D(P)$, how do we show that the spectrum of $P$ consists on only of countably many eigenvalues? I am not sure what techniques are crucial here, but I know that compactness, and kind of Fredholm-alternative, will be relevant. I am familiar with basic theory of unbounded operators on a Hilbert space (for instance, from Reed-Simon vol. 1) but I do not know Sturm-Liouville theory.
Hints or solutions are greatly appreciated!