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?

  • $\begingroup$ 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. $\endgroup$ Feb 7, 2019 at 22:57
  • $\begingroup$ 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 $\lambda_t^0$? $\endgroup$
    – julian
    Feb 7, 2019 at 23:37

1 Answer 1


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) $p(x)$.

A well-known Variation Principle says that when the stiffness increases, while the mass is constant, all frequencies increase. The best reference is

MR1908601 Gantmacher, F. P.; Krein, M. G. Oscillation matrices and kernels and small vibrations of mechanical systems. Revised edition. Translation based on the 1941 Russian original. Edited and with a preface by Alex Eremenko. AMS Chelsea Publishing, Providence, RI, 2002.

  • $\begingroup$ Nice heuristics! $\endgroup$ Jul 8, 2019 at 9:50

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