On domain properties of Bessel-type operators

Motivated by a recent study of Bessel operators in connection with refinement Hardy's inequality involving $ 1/\sin^2(x) on the finite interval (0, \pi) $, we now take closer look at underlying Bessel-type more general inverse square singularities endpoints. More precisely, consider quadratic forms and operator realizations L^2((a, b); dx) associated differential expressions form \begin{document}$ \omega_{s_a} = - \frac{d^2}{dx^2} + \frac{s_a^2 (1/4)}{(x-a)^2}, \quad s_a \in {\mathbb{R}}, \; x (a, b), $\end{document} and$ \begin{align*} \tau_{s_a, s_b} (1/4)}{(x-a)^2} \frac{s_b^2 (1/4)}{(x-b)^2} q(x), & \\ s_a, s_b [0, \infty), q L^{\infty}((a, dx), \text{ real-valued a.e. } b) , \end{align*} $where \subset {\mathbb{R}} is bounded interval.As an explicit illustration describe Krein–von Neumann extension minimal corresponding to $.