On the absolute continuity of the spectrum of Sturm–Liouville operator with matrix singular coefficients

Ya. I. Granovskii^ab, M. M. Malamud<sup>c

a</sup> Donetsk National Technical University
^b Institute of Applied Mathematics and Mechanics, Donetsk
^c Saint Petersburg State University

Abstract: In the present work the spectral structure of realizations on the half-line of a matrix three-term Sturm–Liouville operator
\begin{equation*} \mathcal{L}(P,Q,R)y:=R^{-1}(x)\bigl(-(P(x)y')'+Q(x)y\bigr), y=(y_1,\ldots,y_m)^{\top}, \end{equation*}
with singular potential $Q( \cdot ) = Q( \cdot )^*$ on the half-line is investigated. It is shown that under certain conditions on the coefficients $P( \cdot )$ and $R( \cdot )$ depended from the small parameter $\varepsilon$, the Dirichlet realization $L^D$ (and other self-adjoint realizations) in the case of $Q( \cdot )\in W^{-1,1}(\mathbb{R}_+;\mathbb{C}^{m\times m})$ has Lebesgue non-negative spectrum with constant multiplicity $m$.

Key words and phrases: Schrödinger operators, singular potentials, regularization, boundary triplets, Weyl functions, absolutely continuous spectrum.

UDC: 517.984.4

Received: 25.09.2025