Eigenvalue Ratios and Gaps of Sturm–Liouville Problems

In this paper, we will study the eigenvalue ratios $\frac{\lambda_{n}}{\lambda_{m}}$ and the eigenvalue gaps $\lambda_{n} - \lambda_{m}$ of the Sturm–Liouville problem \[ -(p(x) y')' + q(x)y = \lambda w(x)y. \] We prove that if $q \leq 0$ and $\theta'' \geq 0$, then $\frac{\lambda_{n}}{\lambda_{m}} \leq \big( \frac{n}{m} \big)^{2}$ and if $q \geq 0$ and $\theta'' \leq 0$, then $\frac{\lambda_{n}}{\lambda_{m}} \geq \big( \big\lfloor \frac{n}{m} \big\rfloor \big)^{2}$. In the case where $q \equiv 0$ and $p = 1$, we prove that if $4ww'' \geq 5(w')^{2}$ then $\lambda_{n} - \lambda_{m} \leq \big[ \big( \frac{n}{m} \big)^{2} - 1 \big] \frac{(m\pi)^{2}}{w_{m}}$ and if $4ww'' \leq 5(w')^{2}$, then $\lambda_{n} - \lambda_{m} \geq \big[ \big( \big\lfloor \frac{n}{m} \big\rfloor \big)^{2} - 1 \big] \frac{(m\pi)^{2}}{w_{M}}$, where $\theta(x) = (p(x) w(x))^{1/4}$, $w_{m} = \inf_{x \in [0,1]} w(x)$ and $w_{M} = \max_{w \in [0,1]} w(x)$.