I have a Sturm-Liouville problem in the following form: $$(py')'-qy=-\lambda wy$$ defined in $x \in [0,L]$ and $L<\infty$ with periodic boundary condition. where in my case, $q=0$ and $p=w$ are piecewise constant functions taking the values either $1$ or a small value $\epsilon \ge 0$ in every subinterval.
When $\epsilon>0$ I can derive valid eigenvalue/eigenfunctions without any problem; i.e. the eigenvalues multiplicity is at most 2 and the eigenfunctions can be orthogonalized e.g. using Gram-Schmidt process (since they are not necessarily orthogonal).
However I had noticed as $\epsilon \to 0$, some of the consecutive eigenvalues converge to a common value and when I choose exactly $\epsilon=0$ (the ideal case), they all combine into one value with a multiplicity of more than 2. whereas I read somewhere that since the differential equation is second order, the multiplicity should be no more than 2. Next I tried to employ the Gram-Schmidt algorithm to orthogonalize the derived eigenfunctions, but it failed to yield an orthogonal set of eigenfunctions and even I checked the solutions had become invalidated.
What is the problem? Is it now singular with $\epsilon=0$? Is it even possible with multiplicity of more than 2? Does orthogonalization have some constraints?
The classical Sturm-Liouville problem on a finite interval $[a,b]$ involves the operator $$ Lf = \frac{1}{w}\left[-\frac{d}{dx}\left(p\frac{df}{dx}\right)+qf\right] $$ where $p > 0$ on $(a,b)$ and is continuously differentiable on $[a,b]$, where $w$ is positive and piece-wise continuous on $[a,b]$, and where $q$ is piece-wise continuous on $[a,b]$. Conditions can be relaxed on $p,q,w$, but the conditions above are typical for classical cases. If $p$ is strictly positive on $[a,b]$, then the problem is regular. If $p$ vanishes at $a$ and/or $b$, then the operator is considered to be singular. If you want $p$ to be allowed to vanish at an interior point $c\in(a,b)$, then you'll typically need to consider separate problems on $(a,c)$ and $(c,b)$ and piece the solutions together at $c$.
With this operator $L$, the Sturm-Liouville problem is cast in the setting of weighted space $L^2_w[a,b]$, where the inner product $\langle\cdot,\cdot\rangle_w$ is $$ \langle f,g\rangle_w = \int_a^b f(x)\overline{g(x)}w(x)dx. $$ The classical domain of $L$ consists of absolutely continuous functions $f$ on $(a,b)$. Endpoint conditions naturally arise in the context of looking at the symmetry of the operator $L$: $$ \langle Lf,g\rangle_w-\langle f,Lg\rangle_w =\int_a^b\left\{-(pf')'+qf\right\}\overline{g}-f\left\{-(p\overline{g}')'+q\overline{g}\right\}dx \\ = \int_a^b -(pf')'\overline{g}+f(p\overline{g}')' dx \\ = \int_a^b \frac{d}{dx}\left[-(pf')\overline{g}+f(p\overline{g}')\right]dx \\ = \left.-p\left\{f'\overline{g}-f\overline{g}'\right\}\right|_{x=a}^{b} = W_p(f,g)|_a^b. $$ The weighted Wronskian $W_p$ is defined as $W_p(f,g)=p(f\overline{g}'-f'\overline{g})$.
With these common conventions in mind, your situation where $p$ could vanish somewhere on the interior of $[a,b]$ would not be considered as a Sturm-Liouville problem. You would have to break the problem into multiple Sturm-Liouville problems over adjacent intervals.