子流形上Schrodinger算子特征值的泛不等式
Universal inequalities for the eigenvalues of Schrodinger operators on submanifolds
摘要：
我们在欧几里得空间、球面、实、复和四元数射影空间的紧致子流形（可能有非空边界）上建立了薛定谔算子的特征值不等式，这些不等式与Payne、Pólya、Weinberger和Yang关于欧几里得整环上拉普拉斯算子的不等式有关，但它们以显式的方式依赖于中曲率。在后面的章节中，我们证明了齐次黎曼空间上的薛定谔算子的类似结果，更一般地，证明了任何允许特征映射进入球面的黎曼流形上的薛定谔算子的类似结果，以及海森堡群子域上的科恩拉普拉斯算子的类似结果。在这个分析的后果是一个扩展的赖利不等式，界定任何特征值的拉普拉斯算子的中曲率，和频谱标准的浸入式流形在齐性空间。
英文摘要：
We establish inequalities for the eigenvalues of Schrödinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related to inequalities for the Laplacian on Euclidean domains due to Payne, Pólya, and Weinberger and to Yang, but which depend in an explicit way on the mean curvature. In later sections, we prove similar results for Schrödinger operators on homogeneous Riemannian spaces and, more generally, on any Riemannian manifold that admits an eigenmap into a sphere, as well as for the Kohn Laplacian on subdomains of the Heisenberg group. Among the consequences of this analysis are an extension of Reilly's inequality, bounding any eigenvalue of the Laplacian in terms of the mean curvature, and spectral criteria for the immersibility of manifolds in homogeneous spaces.
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