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\ma...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | deu |
| Published: |
Ivan Franko National University of Lviv
2023-09-01
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| Series: | Математичні Студії |
| Subjects: | |
| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/433 |
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| Summary: | 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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| ISSN: | 1027-4634 2411-0620 |