The Jacobi operator and its Donoghue $m$-functions
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| Format: | Preprint |
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2021
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| _version_ | 1866917735554875392 |
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| author | Gesztesy, Fritz Piorkowski, Mateusz Stanfill, Jonathan |
| author_facet | Gesztesy, Fritz Piorkowski, Mateusz Stanfill, Jonathan |
| contents | In this paper we construct Donoghue $m$-functions for the Jacobi differential operator in $L^2\big((-1,1); (1-x)^α (1+x)^β dx\big)$, associated to the differential expression \begin{align*} \begin{split} τ_{α,β} = - (1-x)^{-α} (1+x)^{-β}(d/dx) \big((1-x)^{α+ 1}(1+x)^{β+ 1}\big) (d/dx),& \\ x \in (-1,1), \; α, β\in \mathbb{R}, \end{split} \end{align*} whenever at least one endpoint, $x=\pm 1$, is in the limit circle case. In doing so, we provide a full treatment of the Jacobi operator's $m$-functions corresponding to coupled boundary conditions whenever both endpoints are in the limit circle case, a topic not covered in the literature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_15913 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The Jacobi operator and its Donoghue $m$-functions Gesztesy, Fritz Piorkowski, Mateusz Stanfill, Jonathan Classical Analysis and ODEs Primary: 34B20, 34B24, 34L05, Secondary: 47A10, 47E05 In this paper we construct Donoghue $m$-functions for the Jacobi differential operator in $L^2\big((-1,1); (1-x)^α (1+x)^β dx\big)$, associated to the differential expression \begin{align*} \begin{split} τ_{α,β} = - (1-x)^{-α} (1+x)^{-β}(d/dx) \big((1-x)^{α+ 1}(1+x)^{β+ 1}\big) (d/dx),& \\ x \in (-1,1), \; α, β\in \mathbb{R}, \end{split} \end{align*} whenever at least one endpoint, $x=\pm 1$, is in the limit circle case. In doing so, we provide a full treatment of the Jacobi operator's $m$-functions corresponding to coupled boundary conditions whenever both endpoints are in the limit circle case, a topic not covered in the literature. |
| title | The Jacobi operator and its Donoghue $m$-functions |
| topic | Classical Analysis and ODEs Primary: 34B20, 34B24, 34L05, Secondary: 47A10, 47E05 |
| url | https://arxiv.org/abs/2110.15913 |