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引用次数: 0
摘要
本文旨在获得经典斯特姆-利乌维尔算子所有节点位置的最优特征,给定势的\(L^p\)规范与\(1<p <\infty\)。将第 m 个特征函数的第 i 个节点视为势的函数,我们推导出临界方程来确定最小化势,从而使节点最小化。根据临界方程,我们可以得到最小节点的两种等效特征,它们可以写成 4 维或 2 维参数的非线性系统。这些最优表征可以得出节点位置的尖锐下限和上限。
Optimization problems on nodes of Sturm–Liouville operators with $$L^p$$ potentials
The aim of this paper is to obtain the optimal characterizations of locations for all nodes of the classical Sturm-Liouville operators, given the \(L^p\) norms with \(1<p <\infty \) of the potentials. Regarding the ith node of the mth eigenfunction as a functional of the potential, we deduce critical equations to determine the minimizing potential such that the node is minimized. From the critical equations, we obtain two equivalent characterizations of the minimal nodes, which are written as nonlinear systems for 4-dimensional or 2-dimensional parameters. These optimal characterizations can yield the sharp lower and upper bounds for the locations of nodes.
期刊介绍:
Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin.
The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin.
Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.