Singular Sturm-Liouville operators with extreme properties that generate black holes

摘要

This study is motivated by several engineering papers that describe the so-called acoustic black hole; for example, a beam with a monotonically decreasing thickness toward one of the endpoints. It appears that the time of propagation of a signal toward such an endpoint is infinite so that the signal is "trapped." Also, the amplitude of a signal increases with no bound near this point. 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 on a finite interval with the coefficients vanishing at one of the endpoints-that is, with extreme coefficients. Physically, that means that we study a rod instead of a beam. We classify the endpoints, compute the essential spectrum, and determine conditions for the absence of positive eigenvalues of the corresponding self-adjoint extensions. The absence of positive discrete spectrum means, physically, that the rod does not sound on any frequency. Our analysis allows us to precisely describe how the coefficients of the differential operator should vanish to produce the essential spectrum and where it is located. An extensive mathematical literature is devoted to the three-dimensional problems of elasticity in bounded domains with a cusp and similar problems for the bodies with a blunted pick. The engineering papers also study the physical properties of an almost sharp beam; that is, the thickness is decreasing toward a small positive limit. In our paper, the spectral properties of the Sturm-Liouville problem with the coefficients almost vanishing at one of the endpoints are studied. We also 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.