Essential self-adjointness of Schrödinger operators with singular potentials
Essential self-adjointness of Schrödinger operators with singular potentials
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奇异势薛定谔算子的本质自伴性
DOI：
10.1007/bf00249091
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发表时间：
1973
影响因子：
2.5
通讯作者：
B. Simon
中科院分区：
文献类型：
--
作者：
B. Simon
本文研究了l2 （Rm）上的Schr6dinger算子-d+ q，其中q是r上与实值可测函数q (x)相乘的算子。我们证明了a + q本质上是自伴随的，或者是C~（Rm /{o}）上的C~（Rm /{o}）上的C~~ （Rm/{o}）上的C~~紧支持函数上的自伴随。有时我们将分别用C~和C~~表示这些集合。这类算子的自伴随性是一个被广泛研究的问题，但直到最近所有的结果都至少假设q在局部Stummel空间（略弱于q6 (LP), p>= 2, p>= 2）。这比条件qE (L2) 1oe （resp）强得多。qe （L2 (R/{O})）需要在C~（R/{O}）上定义a + q。C ~)。一般来说，如果q~ l2和m> __4（见[4]给出一个明确的例子），a + q将不会在C~上自伴随。然而，我们最近证明了如果q_bbb_0，那么q_el2对于C~ bbb_0上的本质自伴随是充分的。KATO[2]对这一结果进行了扩展，下面我们将使用KATO[2]的一些方法。下面我们证明的两个定理是：
In this note we wish to study Schr6dinger operators-d+ q on L 2 (Rm), where q is the operator of multiplication by a real-valued measurable function, q (x), on R. We show that-A+ q is essentially self-adjoint on either C~(Rm), the C oo functions of compact support, or on C~'(Rm/{o}), the C~~ functions of compact support in Rm/{o}. Upon occasion we shall denote these sets by C~ and C~~ respectively.The self-adjointness of such operators is an extensively studied problem, but until recently all results have at least supposed that q is in a local Stummel space (slightly weaker than q6 (LP) lor with p> m/2, p>= 2). This is considerably stronger than the condition qE (L2) 1oe (resp. qe (L2 (R/{O})) lor needed for-A+ q to be well-defined on C~(resp. C~). In general,-A+ q will not be self-adjoint on C~ if q~ L 2 and m> __4 (see [4] for an explicit example). However, we recently showed that if q_> _ 0, then q eL 2 is sufficient for essential self-adjointness on C~[4]. Extensions of this result have been obtained by KATO [2], some of whose methods we shall use below. The two theorems we prove below are: