Singularly perturbed rank one linear operators

Q3 Mathematics Matematychni Studii Pub Date : 2021-12-26 DOI:10.30970/ms.56.2.162-175
M. Dudkin, O. Dyuzhenkova
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Abstract

The basic principles of the theory of singularly perturbed self-adjoint operatorsare generalized to the case of closed linear operators with non-symmetric perturbation of rank one.Namely, firstly linear closed operators are considered that coincide with each other on a dense set in a Hilbert space.The theory of singularly perturbed self-adjoint operators arose from the need to consider differential expressions in such terms as the Dirac $\delta$-function.Since it is important to consider expressions given not only by symmetric operators, the generalization (transfer) of the basic principles of the theory of singularly perturbed self-adjoint operators in the case of non-symmetric ones is important problem. The main facts of the theory include the definition of a singularly perturbed linear operator and the resolvent formula in the cases of ${\mathcal H}_{-1}$-class and ${\mathcal H}_{-2}$-class.The paper additionally describes the possibility of the appearance a point of the point spectrum and the construction of a perturbation with a predetermined point.In comparison with self-adjoint perturbations, the description of perturbations by non-symmetric terms is unexpected.Namely, in some cases, when the perturbed by a vectors from ${\mathcal H}_{-2}$ operator can be conveniently described by methods of class ${\mathcal H}_{-1}$, that is impossible in the case of symmetric perturbations of a self-adjoint operator. The perturbation of self-adjoint operators in a non-symmetric manner fully fits into the proposed studies.Such operators, for example, generalize models with nonlocal interactions, perturbations of the harmonic oscillator by the $\delta$-potentials, and can be used to study perturbations generated by a delay or an anticipation.
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奇摄动秩一线性算子
将奇异摄动自伴随算子理论的基本原理推广到具有非对称1阶摄动的闭线性算子。即,首先考虑在Hilbert空间的密集集合上相互重合的线性闭算子。奇摄动自伴随算子的理论是由于需要考虑狄拉克函数的微分表达式而产生的。由于不仅要考虑对称算子给出的表达式,所以奇摄动自伴随算子理论基本原理在非对称算子情况下的推广(传递)是一个重要的问题。该理论的主要事实包括奇异摄动线性算子的定义以及${\mathcal H}_{-1}$-类和${\mathcal H}_{-2}$-类的解式。此外,本文还讨论了点谱出现点的可能性和带预定点的微扰的构造。与自伴随微扰相比,用非对称项描述微扰是不可预料的。也就是说,在某些情况下,当一个来自${\mathcal H}_{-2}$算子的向量的摄动可以方便地用${\mathcal H}_{-1}$类的方法来描述时,这在自伴随算子的对称摄动情况下是不可能的。非对称自伴随算子的摄动完全符合本文的研究。这样的算子,例如,推广非局部相互作用的模型,由$\ δ $-势对谐振子的扰动,并可用于研究由延迟或预期产生的扰动。
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来源期刊
Matematychni Studii
Matematychni Studii Mathematics-Mathematics (all)
CiteScore
1.00
自引率
0.00%
发文量
38
期刊介绍: Journal is devoted to research in all fields of mathematics.
期刊最新文献
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