Singular Sturm-Liouville operators with extreme properties that generate black holes. (English) Zbl 1487.34083

The main objective of this paper is the rigorous study of the spectral properties of the Sturm-Liouville problem for a second-order differential operator of the form \[ \ell y := (p(x) y')'+\lambda r(x) y =0, \qquad\qquad\qquad\qquad\qquad\qquad (*) \] on the interval \((0,1]\), where \(\lambda=\omega^{2}\) is the spectral parameter, \(p(x)=p_{0}(x) x^{\alpha}\), \(r(x)=r_{0}(x) x^{\beta}\), \(p_{0},r_{0}\) are continuous positive functions on [0,1] and \(\alpha \geq \beta +2\).
Physically, the authors investigate a physical subject associated with \((*)\), so-called acoustic black hole, and study a rod instead of a beam. They are interested in the following phenomena.
Phenomenon I. If \(\int_{0}^{1} \sqrt{\frac{r(x)}{p(x)}} dx=\infty\), then the wave that starts at \(x=1\) will never reach the other, so that the rod represents a black hole.
Phenomenon II. If the Sturm-Liouville problem for the differential equation \((*)\) does not have a positive discrete spectrum, then the rod will not sound on any frequency \(\omega\).
In this paper, the spectral properties of the Sturm-Liouville problem with the coefficients almost vanishing at one of the endpoints are studied. First, the authors describe physical models that motivate their studies, including two models suggested by Mironov, and describe the acoustic black hole phenomena for these models. They summarize the known rigorous spectral results for a plate and beam with a sharp edge, and review some results devoted to the three-dimensional elasticity problems with cusps. Then, they use the WKB method and present the approximate solutions of \((*)\), and show that, for large values of the spectral parameter, the approximation to the solution satisfies the properties found by engineers in their models; that is, the time of propagation toward the sharp end is infinite and the amplitude near that end increases with no bound. They classify the endpoints, compute the essential spectrum, and determine conditions for the absence of positive eigenvalues of the corresponding self-adjoint extensions. Also, they discuss the connection between the spectral properties of the Sturm-Liouville problem with positive coefficients and the original, singular Sturm-Liouville problem.


34B24 Sturm-Liouville theory
34L05 General spectral theory of ordinary differential operators
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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