The Boundary Value Problem

A solution $u(x,\lambda)$ of the S-L d.e. for $a\le x\le b$ will be an eigenfunction of the regular S-L boundary value problem if and only if the corresponding phase, obtained from the Prüfer d.e.

$$\frac{d\theta}{dx} = (\lambda\rho - q)\sin^2\theta +\frac{1}{p}\cos^2\theta\,,$$

satisfies the corresponding end point conditions

$$\theta (a,\lambda )=\gamma~\quad\textrm{and}~\quad \theta (b,\lambda )=\delta +n\pi~~\quad~~n=0,1,\dots$$

with $0\le\gamma <\pi$ and $0<\delta\le\pi$ .

Note that any $\lambda$ for which these endpoint conditions hold is an eigenvalue of the regular S-L problem, and conversely, that an eigenvalue of this S-L problem will yield a phase function whenever it satisfies the required end point conditions for some $n=0,1,2,\dots$ .

The question now is: Does there exist a $\lambda$ which guarantees that the two end conditions are satisfied for every $n=0,1,2,\dots$ ?