In this paper, we consider a random one-dimensional Schrödinger operator Hω on the half line which has, as its potential term, Gaussian white noise multiplied by a decaying factor. Although the potential term is not an ordinary function, but a distribution, it is possible to realize Hω as a symmetric operator in L2([0, ∞); dt) as was pointed out by the present author [Minami, Lect. Notes Math. 1299, 298 (1986)], and it will be shown that Hω is actually self-adjoint with probability one. When the white noise in Hω is replaced by random functions of a specific type, [Kotani and Ushiroya, Commun. Math. Phys. 115, 247 (1988)] made a precise analysis of the positive part of the spectrum. According to them, if the decaying factor is not square integrable, the positive part of the spectrum typically consists of singular continuous and dense pure-point parts, which are separated by a threshold number. On the other hand, the positive part of the spectrum is purely absolutely continuous when the decaying factor is square integrable. In this work, we shall focus on the case of square integrable decaying factor, and prove the absolute continuity of the positive part of the spectrum of Hω. We shall further prove that the negative part of the spectrum of Hω is discrete, with no accumulation points other than 0.

中文翻译：

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具有衰减白噪声势的一维薛定谔算子

在本文中，我们考虑半线上的随机一维薛定谔算子 Hω，其潜在项是高斯白噪声乘以衰减因子。尽管势项不是普通函数，而是分布，但可以将 Hω 实现为 L2([0, ∞); 中的对称算子； dt）正如本作者所指出的[Minami，Lect。数学笔记。 1299, 298 (1986)]，并且将证明 Hω 实际上是自伴的，概率为 1。当 Hω 中的白噪声被特定类型的随机函数取代时，[Kotani 和 Ushiroya, Commun.数学。物理。 115, 247 (1988)]对光谱的正部分进行了精确分析。根据他们的说法，如果衰减因子不可平方可积，则频谱的正部分通常由奇异连续和密集的纯点部分组成，这些部分由阈值数分隔。另一方面，当衰减因子是平方可积时，频谱的正部分是完全绝对连续的。在这项工作中，我们将重点关注平方可积衰减因子的情况，并证明Hω频谱的正部分的绝对连续性。我们将进一步证明Hω频谱的负数部分是离散的，除了0之外没有其他累积点。