In this paper we study spectral properties of a threedimensional Schrödinger operator −∆ + V with a potential V given, modulo rapidly decaying terms, by a function of the distance of x ∈ R to an infinite conical hypersurface with a smooth cross-section. As a main result we show that there are infinitely many discrete eigenvalues accumulating at the bottom of the essential spectrum which itself is identified as the ground-state energy of a certain one-dimensional operator. Most importantly, based on a result of Kirsch and Simon we are able to establish the asymptotic behavior of the eigenvalue counting function using an explicit spectral-geometric quantity associated with the cross-section. This shows a universal character of some previous results on conical layers and δ-potentials created by conical surfaces.

中文翻译：

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电位集中在圆锥面附近的薛定谔算符的离散谱

在本文中，我们研究了三维薛定谔算子 -Δ + V 的光谱特性，其中给定了势 V，模快速衰减项，通过 x ∈ R 到具有光滑横截面的无限圆锥超曲面的距离的函数。作为主要结果，我们表明在本质谱的底部累积了无限多个离散特征值，本质谱本身被识别为某个一维算子的基态能量。最重要的是，基于 Kirsch 和 Simon 的结果，我们能够使用与横截面相关的显式谱几何量来建立特征值计数函数的渐近行为。这显示了先前关于锥形层和锥形表面产生的 δ 势的一些结果的普遍特征。