My questions comes from reading the book The Spectral Theory of Periodic Differential Equations by Eastham.
Consider the following differential equation:
$[p(x)y'(x)]' + [\lambda s(x) + q(x)]y(x) = 0$
where $\lambda \in \mathbb{R}$, $p(x), s(x), q(x)$ are real valued, piece-wise continuous, and with period $a$. Also we assume that $s(x) \geq s>0$ and that $p(x)$ is never zero.
Now we can consider the periodic problem with the initial condtions that $y(a) = y(0), y'(a) = y'(0)$.
This problem is self adjoint, and the book states that there are an infinite countable eigenvalues, which can be ordered as such:
$\lambda_0 \leq \lambda_1 \leq \lambda_2...$
Question 1:
Is this a complete characterization of the eigenvalues? That is, will every value of $\lambda$ for which there is a solution to the above equation with the above initial conditions be counted within that countable set?
Further down the book, he gives this definition:
The above differential equation is called oscillatory if it has a non-trivial solution with an infinite number of roots in $\mathbb{R}$.
Then this theorem follows:
If $\lambda > \lambda_0$ (from above), then the above equation is oscillatory.
Question 2:
How do I know that for every $\lambda$ there is a non-trivial solution to the equation? Can that equation be solved for any real valued $\lambda$ without specifying initial conditions?