变换算子为阻抗Sturm-Liouville算子

Q3 Mathematics
M. Kazanivskiy, Ya. Mykytyuk, N. Sushchyk
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引用次数: 0

摘要

在Hilbert空间$H:=L_2(\mathbb{R})$中,我们考虑由微分表达式$ -p\frac{d}{dx}{\frac1{p^2}}\frac{d}{dx}p$产生的阻抗Sturm—Liouville算子$T:H\to H$,其中函数$p:\mathbb{R}\to\mathbb{R}_+$在$\mathbb{R}$和$\inf_{x\in\mathbb{R}} p(x)>0$上有界变化。研究了算子$T$的变换算子的存在性及其性质。
本文通过$p_\mu(x):= e^{\mu([x,\infty))}, x\in\mathbb{R}.$提出了阻抗函数p用实值有界测度$\mu\in \boldsymbol M$有效参数化的方法。对于测度$\mu\in \boldsymbol M$,我们建立了Sturm—Liouville算子$T_\mu$的变换算子的存在性,该变换算子由函数$p_\mu$构造。证明了算子$T_\mu$对$\mu$的连续依赖。因此,我们推导出运算符$T_\mu$与运算符$T_0:=-d^2/dx^2$是一元等价的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Transformation operators for impedance Sturm–Liouville operators on the line
In the Hilbert space $H:=L_2(\mathbb{R})$, we consider the impedance Sturm--Liouville operator $T:H\to H$ generated by the differential expression $ -p\frac{d}{dx}{\frac1{p^2}}\frac{d}{dx}p$, where the function $p:\mathbb{R}\to\mathbb{R}_+$ is of bounded variation on $\mathbb{R}$ and $\inf_{x\in\mathbb{R}} p(x)>0$. Existence of the transformation operator for the operator $T$ and its properties are studied. In the paper, we suggest an efficient parametrization of the impedance function p in term of a real-valued bounded measure $\mu\in \boldsymbol M$ via$p_\mu(x):= e^{\mu([x,\infty))}, x\in\mathbb{R}.$For a measure $\mu\in \boldsymbol M$, we establish existence of the transformation operator for the Sturm--Liouville operator $T_\mu$, which is constructed with the function $p_\mu$. Continuous dependence of the operator $T_\mu$ on $\mu$ is also proved. As a consequence, we deduce that the operator $T_\mu$ is unitarily equivalent to the operator $T_0:=-d^2/dx^2$.
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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.
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