Quasiregularity of a singular differential operator (Q1803064)

The two-dimensional differential expression \[ \ell(y)= (P_0 (x) y')'+ P_1 (x)y, \qquad 0\leq x< \infty, \tag{1} \] \[ P_0 (x)= \begin{pmatrix} 0 &p_0 (x)\\ p_0 (x) &0\end{pmatrix}, \qquad P_1 (x)= \begin{pmatrix} p(x) &q(x)\\ q(x) &r(x) \end{pmatrix} \] with Lebesgue measurable elements, \(p_0\) invertible, is considered. \(L\) denotes the minimal, closed and symmetric operator generated by \(\ell(y)\) in the complex Hilbert space \(L^2_2 (0, \infty)\). It is known that the deficiency index of \(L\) is 2, 3 or 4. When it is equal to 4, then \(L\) is called quasiregular. In the paper some necessary and sufficient conditions for quasiregularity are demonstrated. These theorems lead to conditions under which \(L\) is not a quasiregular operator.