Sturm-Liouville operators with $W^{-1,1}$-matrix potentials

Ya. I. Granovskiy ; M. M. Malamud

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 52, Tome 516 (2022), pp. 20-39

Résumé

In the present work the spectral structure of realizations 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 and line is investigated. It is shown that under certain conditions on the coefficients $P( \cdot )$ and $R( \cdot )$ 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 of constant multiplicity $m$. In particular, Schrödinger operator with matrix potential $Q( \cdot )\in W^{-1,1}(\mathbb{R}_+;\mathbb{C}^{m\times m})$ has Lebesgue non-negative spectrum of 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 $L := L_{\min }$ in $L^2(\mathbb{R};R;\mathbb{C}^m)$ is self-adjoint, then the non-negative spectrum of the operator $L$ is Lebesgue of constant multiplicity $2m$ whenever $Q( \cdot )\mathbf{1}_{\mathbb{R}_+}(\cdot) \in W^{-1,1}(\mathbb{R}_+;\mathbb{C}^{m\times m})$. In particular, if the minimal Schrödinger operator $\mathbf{H}$ 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, $\mathbf{H} = \mathbf{H}^*$, then its non-negative spectrum is Lebesgue one of constant multiplicity $2m$ whenever $Q_1( \cdot )\mathbf{1}_{\mathbb{R}_+}\in L^1(\mathbb{R}_+;\mathbb{C}^{m\times m})$ and $\sum\limits_{k=1}^{\infty}|\alpha_k|\infty$.