On domain properties of Bessel-type operators
| Autor: | Gesztesy, Fritz, Pang, Michael M. H., Stanfill, Jonathan |
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| Rok vydání: | 2021 |
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| Zdroj: | Discrete and Continuous Dynamical Systems - S, 2024, 17(5&6): 1911-1946 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.3934/dcdss.2022201 |
| Popis: | 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 $\tau_{s_a,s_b}$. Comment: 37 pages, references updated |
| Databáze: | arXiv |
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