Singular fourth-order Sturm–Liouville operators and acoustic black holes

Abstract

We derive conditions for a one-term fourth-order Sturm–Liouville operator on a finite interval with one singular endpoint to have essential spectrum equal to $[0,\infty )$ or $\varnothing $. Of particular usefulness are Kummer–Liouville transformations which have been a valuable tool in the study of second-order equations. Applications to a mechanical beam with a thickness tapering to zero at one of the endpoints are considered. When the thickness $2h$ satisfies $c_1x^{\nu }\leq h(x)\leq c_2x^{\nu }$, we show that the essential spectrum is empty if and only if $\nu < 2$. As a final application, we consider a tapered beam on a Winkler foundation and derive sufficient conditions on the beam thickness and the foundational rigidity to guarantee the essential spectrum is equal to $[0,\infty )$.