Uniqueness of the self-adjoint extension of singular elliptic differential operators
Uniqueness of the self-adjoint extension of singular elliptic differential operators
复制标题
奇异椭圆微分算子自伴扩张的唯一性
DOI：
10.1007/bf00253334
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发表时间：
1962
影响因子：
2.5
通讯作者：
Tosio Kato
中科院分区：
文献类型：
--
作者：
Teruo Ikebe;Tosio Kato
本文考虑形式为(T)Tu=~.[IOI+bi(X)]AIk(X)[Iok+bk(X)]u+Q(X)u的二阶椭圆型微分算子Xer“S，其中AIk(X)是实数，AIk(X)=aki(X)，ai=a/~x/，x=(x1……)X，，)和i=V--t。T通常为SD/-adioi~，但是.它在两个方面是奇异的：第一，下标域是整个R“，其中椭圆性不需要是一致的；第二，允许系数Q(X)具有相当强的奇异性，尽管假设其他系数aik(X)，bi(X)是光滑函数。我们的主要目的是给出一个非常一般的充分条件。形式微分算子T决定了Hilhert空间LS(Rm)中唯一的自伴算子，形式(T)的微分算子在应用中经常出现。特别地，(T)的一个特例是由S粒子组成的量子力学系统的薛定谔算符，该系统通过静电势相互作用，并受到外加静电场和磁场的作用。在这种情况下，m-3 S，我们可以假设aik(X)=ik(克隆尼克符号)；bi(Z)表示描述磁场的外部矢量势的分量，q(X)。表示外部静电势以及粒子之间的相互作用势。这些相互作用势通常具有很强的奇异性；这就是为什么应该允许Q(X)是高度奇异性的原因。物理要求这样的算符T实际上以唯一的方式确定自伴算符。因此，对于形式为(T)的算子T确定唯一的自伴算子具有重要的实际意义。这个问题可以表述如下。设T具有区域C~(R上具有紧载体的所有C类函数的集合)对T的限制。在什么条件下，To本质上是L~(R‘~)中的一个算子？在作者[4]的一篇较早的论文中，这个问题是在AIK(X)~Oik，Bj(X)=O和Q(X)的特殊情况下考虑的，其中Q(X)具有适合于量子力学要求的形式(库仑势略为1。这个公式不是Ye。挺完整的，对于Tu需要，不属于LI(R)的所有UEC~。关于Q(X)的一些假设是定义在整个C~上具有L I中的值的必要条件。
In the present paper we consider~ second-order, elliptic differential operator of the form (t) Tu=~.[i Oi+ bi (x)] aik (X)[i O k+ bk (x)] u+ q (x) u for xER" s, where aik (x) are real, aik (x)= aki (x), ai= a/~ x/, x=(x 1..... x,,) and i= V--t. T is/ormally sd/-adioi~, but. it is Singular in two respects: first, the underlying domain is the whole R", where the elliptieity need not be uniform, and, second, the coefficient q (x) is allowed to have rather strong singularities, though the other coefficients aik (x), bi (x) are assumed to be smooth functions. Our main object is to give a very general sufficient condition under which the. formal differential operator T. determines a unique self-adjoint operator in the Hilhert space Ls (Rm).Differential operators of the form (t)" appear frequently in applications. In particular, a special case of (t) is the Schr6dinger operator for a quantummechanical system consisting of s particles, interacting with each other through a static potential and subjected to external electrostatic and magnetic fields. In this case m----3 s, and we may assume that aik (X)=~ ik (Kroneeker's symbol); the bi (z) represent the components of the external vector potential describing the magnetic field, and q (x). represents the external electrostatic potential as well as the potentials of interaction between the particles. These interaction potentials usually have strong singularities; this is the reason why q (x) should be allowed to be highly singular. It is required by physics that such an operator T in fact determine a self-adjoint operator in a unique fashion. Thus it is of practical importance to have a useful criterion for an operator T of the form (t) to determine a unique self-adjoint operator. This problem may be formulated as follows. Let TO be the restriction of T with domain C~(the set of all functions of class C on R with compact carriers). Under what conditions is T o essentially sd/-ad] oi~ as an operator in L~(R'~)? x In an earlier paper of one of the authors [4], this problem was considered in the special case in which aik (X)~ Oik, bj (x)= O and q (x) has a form adapted to the requirement of quantum mechanics (Coulomb potentials in a slightly 1 This formulation is not ye. quite complete, for Tu need, not belong to LI (R) for all ueC~. Some assumptions on q (x) are necessary in order that T o be defined on the whole C~ with values in L I.