Finer limit circle/limit point classification for Sturm-Liouville operators
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916314059112448 |
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| author | Piorkowski, Mateusz Stanfill, Jonathan |
| author_facet | Piorkowski, Mateusz Stanfill, Jonathan |
| contents | In this paper we introduce an index $\ell_c \in \mathbb{N}_0 \cup \lbrace \infty \rbrace$ which we call the `regularization index' associated to the endpoints, $c\in\{a,b\}$, of nonoscillatory Sturm-Liouville differential expressions with trace class resolvents. This notion extends the limit circle/limit point dichotomy in the sense that $\ell_c~=~0$ at some endpoint if and only if the expression is in the limit circle case. In the limit point case $\ell_c>0$, a natural interpretation in terms of iterated Darboux transforms is provided. We also show stability of the index $\ell_c$ for a suitable class of perturbations, extending earlier work on perturbations of spherical Schrödinger operators to the case of general three-coefficient Sturm-Liouville operators. We demonstrate our results by considering a variety of examples including generalized Bessel operators, Jacobi differential operators, and Schrödinger operators on the half-line with power potentials. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_04847 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finer limit circle/limit point classification for Sturm-Liouville operators Piorkowski, Mateusz Stanfill, Jonathan Spectral Theory Classical Analysis and ODEs 34B20, 34B24, 34L05, 47A10 (Primary) 34B30, 34C10, 34C11 (Secondary) In this paper we introduce an index $\ell_c \in \mathbb{N}_0 \cup \lbrace \infty \rbrace$ which we call the `regularization index' associated to the endpoints, $c\in\{a,b\}$, of nonoscillatory Sturm-Liouville differential expressions with trace class resolvents. This notion extends the limit circle/limit point dichotomy in the sense that $\ell_c~=~0$ at some endpoint if and only if the expression is in the limit circle case. In the limit point case $\ell_c>0$, a natural interpretation in terms of iterated Darboux transforms is provided. We also show stability of the index $\ell_c$ for a suitable class of perturbations, extending earlier work on perturbations of spherical Schrödinger operators to the case of general three-coefficient Sturm-Liouville operators. We demonstrate our results by considering a variety of examples including generalized Bessel operators, Jacobi differential operators, and Schrödinger operators on the half-line with power potentials. |
| title | Finer limit circle/limit point classification for Sturm-Liouville operators |
| topic | Spectral Theory Classical Analysis and ODEs 34B20, 34B24, 34L05, 47A10 (Primary) 34B30, 34C10, 34C11 (Secondary) |
| url | https://arxiv.org/abs/2407.04847 |