Let $f \in C^{\infty}([0, 2\pi])$ be a smooth function and consider the following periodic Sturm-Liouville problem:

$$\begin{cases} u''(x) + f(x)u(x) = - \lambda u(x) \\
u(0) = u(2\pi) \\
u'(0) = u'(2\pi)
\end{cases}$$

Is it true that there are only countably many eigenvalues for this problem?

I would also like to know if there are mild sufficient conditions on $f$ so that the above problem has a negative simple eigenvalue $\lambda$.