Transformation operators for impedance Sturm–Liouville operators on the line

  • M. Kazanivskiy Ivan Franko National University of Lviv
  • Ya. Mykytyuk Ivan Franko National University of Lviv
  • N. Sushchyk Ivan Franko National University of Lviv

Анотація

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 $T_0:=-d^2/dx^2$.

Біографії авторів

M. Kazanivskiy, Ivan Franko National University of Lviv

Ivan Franko National University of Lviv

Ya. Mykytyuk, Ivan Franko National University of Lviv

Ivan Franko National University of Lviv

N. Sushchyk, Ivan Franko National University of Lviv

Ivan Franko National University of Lviv

Посилання

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Опубліковано
2023-09-22
Як цитувати
Kazanivskiy, M., Mykytyuk, Y., & Sushchyk, N. (2023). Transformation operators for impedance Sturm–Liouville operators on the line. Математичні студії, 60(1), 79-98. https://doi.org/10.30970/ms.60.1.79-98
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