# Monotonicity/Scaling of Sturm-Liouville Eigenvalues

Consider the regular Sturm-Liouville eigenvalue equation $$\frac{d}{dx}(p_t(x)f^\prime(x))=\lambda_t f(x)$$ for $$p_t\in\mathcal{C}^\infty([0,1])$$ with Dirichlet boundary conditions on $$[0,1]$$. Here $$t\geq 0$$ is a parameter and let's say $$t\mapsto p_t(x)$$ is smooth for all $$x\in[0,1]$$ and $$p_0=1$$. Denote the smallest eigenvalue $$\lambda_t$$ by $$\lambda_t^0$$. So $$\lambda_0^0=\pi^2$$.

Can anything be said about the mapping $$t\mapsto \lambda_t^0$$? E.g. is it monotone in $$t$$ (likely to depend on $$p_t$$), what's its scaling behavior, or is it smooth?

• The setup is too general for monotonicity/scaling since you can always destroy such properties by reparametrizing. Smoothness is no big problem I think, should follow quickly from the implicit function theorem. Feb 7, 2019 at 22:57
• Thank you for the comment. I suspected that there is no general statement possible. Do you know any references which work out monotonicity for a particular example? Is there anything known about the relationship between $p_t$ and the eigenvalue $\lambda_t^0$? Feb 7, 2019 at 23:37

The assumption of smoothness of $$t\mapsto p_t$$ cannot imply monotonicity, as @Christian Remling noted in his comment.
But if you assume that $$t\mapsto p_t$$ is monotone (pointwise) then $$t\mapsto\lambda_t$$ is also monotone.
Sturm-Liouville eigenvalue problems have the following mechanical interpretation. Your eiegnalue problem describes the fundamental frequency $$\lambda$$ of a string with fixed ends, of constant density, and variable stiffness (Hooke's const) $$p(x)$$.