On domain properties of Bessel-type operators
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866909270870589440 |
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| author | Gesztesy, Fritz Pang, Michael M. H. Stanfill, Jonathan |
| author_facet | Gesztesy, Fritz Pang, Michael M. H. Stanfill, Jonathan |
| contents | 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 $(0,π)$, 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 \[ ω_{s_a} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2}, \quad s_a \in \mathbb{R}, \; x \in (a,b), \] and \begin{align*} τ_{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 $ω_{s_a}$ and $τ_{s_a,s_b}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2107_09271 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On domain properties of Bessel-type operators Gesztesy, Fritz Pang, Michael M. H. Stanfill, Jonathan Classical Analysis and ODEs Primary: 26D10, 34A40, 34B20, 34B30, Secondary: 34L10, 34B24, 47A07 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 $(0,π)$, 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 \[ ω_{s_a} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2}, \quad s_a \in \mathbb{R}, \; x \in (a,b), \] and \begin{align*} τ_{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 $ω_{s_a}$ and $τ_{s_a,s_b}$. |
| title | On domain properties of Bessel-type operators |
| topic | Classical Analysis and ODEs Primary: 26D10, 34A40, 34B20, 34B30, Secondary: 34L10, 34B24, 47A07 |
| url | https://arxiv.org/abs/2107.09271 |