Transformation operators for impedance Sturm–Liouville operators on the line

Abstract

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