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.

中文翻译：

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具有产生黑洞的极端性质的奇异 Sturm-Liouville 算子

这项研究的动机是几篇描述所谓的声学黑洞的工程论文。例如，朝向端点之一单调递减厚度的梁。信号向这样的端点传播的时间似乎是无限的，因此信号被“捕获”。此外，信号的幅度在该点附近没有界限地增加。本文的主要目标是严格研究 Sturm-Liouville 问题在有限区间上的二阶微分算子的谱特性，其中系数在端点之一（即具有极值系数）消失。从物理上讲，这意味着我们研究的是杆而不是梁。我们对端点进行分类，计算基本谱，并确定相应自伴随扩展的正特征值不存在的条件。没有正离散频谱意味着，在物理上，棒在任何频率上都不会发声。我们的分析使我们能够精确描述微分算子的系数应该如何消失以产生基本频谱以及它位于何处。大量的数学文献专门研究具有尖头的有界域中的弹性的三维问题以及具有钝尖头的物体的类似问题。工程论文还研究了近乎锐利的光束的物理特性；也就是说，厚度正朝着一个小的正极限减小。在我们的论文中，研究了系数在端点之一几乎为零的 Sturm-Liouville 问题的谱特性。我们还表明，对于较大的光谱参数值，解的近似满足工程师在其模型中发现的属性；也就是说，向尖端传播的时间是无限的，靠近尖端的幅度没有限制地增加。