Let $p$ and $r$ be regular (let say $C^2$) functions on $[0,1]$. Moreover, we assume that $p(t)>0$ for every $t\in[0,1]$.
By Cauchy-Lipschitz theorem, there is a 2-dimensional vector space of solutions of the Sturm-Liouville equation:
$$(px')'+\lambda rx=0$$ where $\lambda>0$. A special case is when $x$ satisfies $x(0)=x(1)=0$. Such a solution is called a Dirichlet solution. The values of $\lambda$ for which a Dirichlet solution does exists are called the Dirichlet eigenvalues of the Sturm Liouville equation. This can be proved that the Dirichlet eigenvalues consists in an increasing sequence $(\mu_k)_{k\in\mathbb{N}}$.
This is proved for example in the book "Theory of ordinary differential equations" of Coddington and Levinson. However, there are other possible boundary conditions. For example, if $p(0)=p(1)$, one can ask for $x$ to satisfy $x(0)=x(1)$ and $x'(0)=x'(1)$. These solutions are called the periodic solutions of the Sturm Liouville equations. The associated eigenvalues are $(\lambda_k)_{k\in\mathbb{N}}$ . We call them the periodic eigenvalues.
By lemma 3.1 in chapter 8 of the book of Coddington and Levinson, we have the following interlacement property: $$\lambda_0\leq\mu_0\leq\lambda_1\leq\mu_1\leq\lambda_2\leq\cdots$$
My problem is that I would like to relax the hypothesis $p(0)=p(1)$ but I still want to have a property of interlacement with respect to the Dirichlet eigenvalues. What is going on if $p(0)\neq p(1)$? Of course, we have to change the boundary conditions.
I have good reasons to think (but it would be long to explain why) that the convenient "pseudo-periodic" boundary condition is $$ (p(0)r(0))x(0)=(p(1)r(1))^{1/4}x(1), (p(0)r(0))^{1/4}x'(0)=(p(1)r(1))^{1/4}x'(1).$$
Do you think that this is correct? If so, do you have a reference for this kind of result?