Eigenfunction expansions associated with singular differential operators
Eigenfunction expansions associated with singular differential operators
复制标题
与奇异微分算子相关的本征函数展开
DOI：
10.1090/s0002-9947-1955-0068701-2
复制
发表时间：
1955
期刊：
影响因子：
--
通讯作者：
J. Elliott
中科院分区：
文献类型：
--
作者：
J. Elliott
在一个区间上 — oo^n^x^r2^oo；假设系数 b(x) 在 (ru r2) 中连续，但不一定有界，并且假设 c(x) 在 [fi, r2] 中连续。我们将对 b(x) 施加其他全局特征的条件。 （参见第 2 节。）受扩散理论应用的启发，我们将考虑巴纳赫空间 C[ri, r2] oi 函数在 [ru r2] 上连续的 (1.1) 和 (1.2) 在 (ri, r2) 上可积函数的空间 L(ri, r2) 中。更具体地说，fi 将被视为从 C[ri, r2] 到其自身的算子，fi* 将被视为从 T.(ri, ri) 到其自身的算子。有关 fi 和 fi* 域的更详细讨论，请参阅§2 和§3 的末尾。我们在这里使用的方法是完全真实的，可以概述如下：我们从（1.1）开始，在类型（3.3）-（3.5）的边界条件下。在我们对 b(x) 的限制下，我们发现边界条件下 fi 的本征函数包含一组正交且完全在某个加权 L2 空间中（§4）。为了弥合希尔伯特空间和巴纳赫空间之间的差距，我们使用半群理论。 Feller [l] 已经表明，对于我们考虑的类型的每组边界条件，都对应一个半群 { Tt}，它在定义 5.1 给出的空间 X 中在 t = 0 处强连续（参见引理 5.1）。我们能够根据 X 中 fi 的本征函数获得 Ttf 的展开（引理 5.2 和 5.3）。这种展开给出了半群的所谓狄利克雷表示（参见 Hille [2, p. 346 以及公式 (21.3.4)-(21.3.6)]）。来自强者
on an interval — oo^n^x^r2^oo; the coefficient b(x) is assumed continuous, but not necessarily bounded, in (ru r2), and c(x) is assumed continuous in [fi, r2]. We shall impose other conditions of a global character on b(x). (See §2.) Motivated by applications to diffusion theory, we shall consider (1.1) in the Banach space C[ri, r2] oi functions continuous on [ru r2] and (1.2) in the space L(ri, r2) of functions integrable on (ri, r2). More specifically, fi will be considered as an operator from C[ri, r2] to itself and fi* as an operator from T.(ri, ri) to itself. For a more detailed discussion of the domains of fi and fi*, see the end of §2 and §3. The method we use here, which is entirely real, may be outlined as follows: we start with (1.1) under boundary conditions of the type (3.3)-(3.5). Under our restrictions on b(x), we find that the eigenfunctions of fi under the boundary conditions contain a set orthonormal and complete in a certain weighted L2 space (§4). To bridge the gap between the Hilbert space and the Banach spaces we use the theory of semi-groups. Feller [l ] has shown that to each set of boundary conditions of the type we consider, there corresponds a semi-group { Tt} which is strongly continuous at t = 0 in the space X given 'by our Definition 5.1 (see our Lemma 5.1). We are able to obtain an expansion for Ttf in terms of the eigenfunctions of fi in X (Lemmas 5.2 and 5.3). This expansion gives the so-called Dirichlet representation of the semi-group (see Hille [2, p. 346 and also formulas (21.3.4)-(21.3.6)]). From the strong