On selfadjoint dilation of the dissipative extension of a direct sum differential operator
The authors consider the differential expression \[ \ell(y):= \frac{1}{w(x)}\left[-y''+\frac{\nu^2-\frac{1}{4}}{x^2}y+q(x)y\right],\quad x\in\Omega:=\Omega_1\cup\Omega_2, \] where \(\Omega_1:=(0; c)\), \(\Omega_2 := (c;\infty)\), \(0 < c < \infty\) and \(0 \leq \nu < 1\). Throughout the paper, it is assumed that the points \(0, c\) and \(\infty\) are singular for the differential expression \(\ell\). Further, it is assumed that \(w\) and \(q\) are real-valued, Lebesgue measurable functions on \(\Omega\), locally integrable functions on \(\Omega_1\) and \(\Omega_2\), and \(w(x) > 0\) for almost all \(x\in\Omega\). The authors describe all maximal dissipative, maximal accretive and selfadjoint extensions of the minimal symmetric operators \(L_0\) corresponding to the differential expression \(\ell\). Further using the equivalence of the Lax-Phillips scattering function and the Sz.-Nagy-Foiaş characteristic function, they show that all eigenfunctions and associated functions of the maximal dissipative extension of \(L_0\) are complete in \(L_{w}^{2}(\Omega)\).
scientific article; zbMATH DE number 6147467
scientific article