The spectral structure of realizations for a matrix three-term Sturm–Liouville operator \(\begin{aligned} \mathcal {L}(P,Q,R)y:=R^{-1}(x)\bigl (-(P(x)y')'+Q(x)y\bigr ), \qquad y=(y_1,\ldots ,y_m)^{\top }, \end{aligned}\) with singular potential \(Q(\,\cdot \,) = Q(\,\cdot \,)^*\) on both the half-line and the line is investigated. It is shown that under certain conditions on the coefficients \(P(\,\cdot \,)\) and \(R(\,\cdot \,)\) , the Dirichlet realization \(L^D\) (as well as other selfadjoint realizations) in the case of \(Q(\,\cdot \,)\in W^{-1,1}(\mathbb {R}_+;\mathbb {C}^{m\times m})\) has Lebesgue nonnegative spectrum with constant multiplicity m. In particular, the Schrödinger operator with matrix potential \(Q(\,\cdot \,)\in W^{-1,1}(\mathbb {R}_+;\mathbb {C}^{m\times m})\) on the half-line \(\mathbb {R}_+\) has Lebesgue spectrum with constant multiplicity m. This result is applied to the Sturm–Liouville expression \(\mathcal {L}(P,Q,R)\) with delta-interactions on the line \(\mathbb {R}\) . It is shown that if the minimal operator \(L:= L_{\text {min}}\) in \(L^2(\mathbb {R};R;\mathbb {C}^m)\) is selfadjoint, then the nonnegative spectrum of L is Lebesgue of constant multiplicity 2m whenever \(Q(\,\cdot \,)\textbf{1}_{\mathbb {R}_+}(\,\cdot \,)\in W^{-1,1}(\mathbb {R}_+;\mathbb {C}^{m\times m})\) . In particular, if the minimal Schrödinger operator \(\textbf{H}\) on the line with potential matrix \(Q(\,\cdot \,)=Q_1(\,\cdot \,)+\sum \limits _{k\in \mathbb {Z}}\alpha _k\delta (\,\cdot \,-x_k)\) is selfadjoint, \(\textbf{H} = \textbf{H}^*\) , then its nonnegative spectrum is Lebesgue with constant multiplicity 2m whenever \(Q_1(\,\cdot \,)\textbf{1}_{\mathbb {R}_+}\in L^1(\mathbb {R}_+;\mathbb {C}^{m\times m})\) and \(\sum \limits _{k=1}^{\infty }|\alpha _k|<\infty \) . Bibliography: 21 titles.
