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Boris P Belinskiy, Don B Hinton, Roger A Nichols, Singular fourth-order Sturm–Liouville operators and acoustic black holes, IMA Journal of Applied Mathematics, Volume 87, Issue 5, October 2022, Pages 804–851, https://doi.org/10.1093/imamat/hxac021
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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 |$[0,\infty )$|.
