TY - JOUR
AU - Kazanivskiy, M.
AU - Mykytyuk, Ya.
AU - Sushchyk, N.
PY - 2023/09/22
Y2 - 2024/09/19
TI - Transformation operators for impedance Sturmâ€“Liouville operators on the line
JF - Matematychni Studii
JA - Mat. Stud.
VL - 60
IS - 1
SE - Articles
DO - 10.30970/ms.60.1.79-98
UR - http://www.matstud.org.ua/ojs/index.php/matstud/article/view/433
SP - 79-98
AB - 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 $T_0:=-d^2/dx^2$.
ER -