Genericity of simple eigenvalues for elliptic PDE’s
Genericity of simple eigenvalues for elliptic PDE’s
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椭圆偏微分方程的简单特征值的通用性
DOI：
10.1090/s0002-9939-1975-0385934-4
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发表时间：
1975
期刊：
影响因子：
--
通讯作者：
J. Albert
中科院分区：
文献类型：
--
作者：
J. Albert
紧致流形上的自伴C'线性椭圆型偏微分算子的谱只包含孤立的本征值，每个本征值具有有限重数。有时这些多重性是无界的;这在应用中出现的问题中很常见，因为通常存在高度对称性。主要定理表明，只有简单的特征值的性质是一般的运营商通过改变一个给定的运营商的零阶部分。这篇文章的目的是提供以下定理的证明。主要定理设M是一个无边界的紧致连通C'流形.设L是M上的自伴C'线性椭圆微分算子.集合Ep ∈ Cw（M）：L + p的所有特征值都是单的}在C '（M）中是剩余的。本文中使用的术语椭圆算子将始终意味着该算子是自伴的、C'和线性的。主要定理可以概括为：几乎所有通过改变给定算子的零阶部分得到的椭圆算子都只有简单的本征值。关于术语，请参阅？1.主要定理首先在[1]中宣布，用于二阶算子，证明是在[2]中给出的;它使用扰动理论。利用横截性理论的一个证明，适用于本征函数满足强唯一连续性的算子。乌伦贝克[6]。该定理在获得本征函数的普适性结果时特别有用（见[1]，[2]，[6]）。1.注释;证明大纲。流形M和算子L将像定理中一样始终固定。关于椭圆算子的标准术语，参见[3]，[4]。编辑于1973年6月4日收到，修订版于1974年1月23日收到。AMS（MOS）主题分类（1970年）。小学35 P05、35 J30;中学47 A55、47 B25、58 G99。
The spectrum of a selfadjoint, C' linear elliptic partial differential operator on a compact manifold contains only isolated eigenvalues, each having finite multiplicity. It is sometimes the case that these multiplicities are unbounded; this is common in problems arising in applications because of the high degree of symmetry usually present. The main theorem shows that the property of having only simple eigenvalues is generic for operators obtained by varying the zeroth order part of a given operator. The purpose of this article is to provide a proof of the following theorem. Main theorem. Let M be a compact, connected C' manifold without boundary. Let L be a sel/adjoint, C' linear elliptic differential operator on M. The set Ep e Cw(M): all eigenvalues of L + p are simple} is residual in C'(M). The term elliptic operator used in this article will always mean the operator is selfadjoint, C' and linear. The main theorem can be summarized by saying that almost all elliptic operators obtained by varying the zeroth order part of a given operator have only simple eigenvalues. For terminology, see ? 1. The main theorem was first announced in [1] for second-order operators, and the proof is the one given in [2]; it uses perturbation theory. A proof using transversality theory, applicable to operators whose eigenfunctions satisfy the strong unique continuation property, has been obtained by K. Uhlenbeck [6]. The theorem is particularly useful in obtaining genericity results about the eigenfunctions (see [1] , [2], [6]). 1. Notation; outline of proof. The manifold M and the operator L will be fixed throughout as in the theorem. For standard terminology on elliptic operators, see [3], [4]. Received by the editors June 4, 1973 and, in revised form, January 23, 1974. AMS (MOS) subject classifications (1970). Primary 35P05, 35J30; Secondary 47A55, 47B25, 58G99.