Quasiregularity conditions for a ''pure imaginary'' singular differential operator of third order (Q1274014)

The author investigates the formally selfadjoint third-order singular differential expression \[ l[y]=-iy'''+{i\over 2}\biggl\{\bigl(p(x) y\bigr)'+p(x)y'\biggr\}, \quad 0\leq x< \infty, \] with pure imaginary coefficients, where \(p(x)\) is a real-valued function on the semiaxis \([0, \infty)\) absolutely continuous on each interval \([0,b]\) \((0<b< \infty)\). He gives sufficient conditions on \(p\) for the operator \({\mathcal L}\) to be quasiregular or, conversely, nonquasiregular. In contrast to other studies the author derives these conditions not by investigating the asymptotic behavior of solutions to the equation \(l[y]=\lambda y\) with the complex parameter \(\lambda\), but by studying properties of the bilinear form \([y,z]= -i(y''\overline z-y'\overline z'+y\overline z'')+ipy\overline z\) associated with \({\mathcal L}\).