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