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On domain properties of Bessel-type operators (2107.09271v2)

Published 20 Jul 2021 in math.CA

Abstract: Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $1/\sin2(x)$ 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 $L2((a,b); dx)$ associated with differential expressions of the form [ \omega_{s_a} = - \frac{d2}{dx2} + \frac{s_a2 - (1/4)}{(x-a)2}, \quad s_a \in \mathbb{R}, \; x \in (a,b), ] and \begin{align*} \tau_{s_a,s_b} = - \frac{d2}{dx2} + \frac{s_a2 - (1/4)}{(x-a)2} + \frac{s_b2 - (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 $\tau_{s_a,s_b}$.

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