Singular Fourth-Order Sturm–Liouville Operators and Acoustic Black Holes

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,infinity] or circle divide. 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(1)x(v) <= h(x) <= c(2)x(v), we show that the essential spectrum is empty if and only if v < 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,infinity] .