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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.00719 |
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| _version_ | 1866917964799803392 |
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| author | Alves, Victor Silva, Guilherme |
| author_facet | Alves, Victor Silva, Guilherme |
| contents | The classical Pólya-Tchebotarev problem, commonly stated as a max-min logarithmic energy problem, asks for finding a compact of minimal capacity in the complex plane which connects a prescribed collection of fixed points. Variants of this problem have found ramifications and applications in the theory of non-hermitian orthogonal polynomials, random matrices, approximation theory, among others. Here we consider an extension of this classical problem, including a semiclassical external field, and enforcing finitely many prescribed collections of points to be connected, possibly also to infinity. Our method is based on Rakhmanov's approach to max-min problems in logarithmic potential theory, utilizes the developed machinery by Martínez-Finkelshtein and Rakhmanov on critical measures, and extends the development of Kuijlaars and the second named author from the context of polynomial external fields to the semiclassical case considered here. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00719 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Pólya-Tchebotarev problem with semiclassical external fields Alves, Victor Silva, Guilherme Classical Analysis and ODEs Complex Variables 31A15 (Primary), 30F30 (Secondary) The classical Pólya-Tchebotarev problem, commonly stated as a max-min logarithmic energy problem, asks for finding a compact of minimal capacity in the complex plane which connects a prescribed collection of fixed points. Variants of this problem have found ramifications and applications in the theory of non-hermitian orthogonal polynomials, random matrices, approximation theory, among others. Here we consider an extension of this classical problem, including a semiclassical external field, and enforcing finitely many prescribed collections of points to be connected, possibly also to infinity. Our method is based on Rakhmanov's approach to max-min problems in logarithmic potential theory, utilizes the developed machinery by Martínez-Finkelshtein and Rakhmanov on critical measures, and extends the development of Kuijlaars and the second named author from the context of polynomial external fields to the semiclassical case considered here. |
| title | The Pólya-Tchebotarev problem with semiclassical external fields |
| topic | Classical Analysis and ODEs Complex Variables 31A15 (Primary), 30F30 (Secondary) |
| url | https://arxiv.org/abs/2403.00719 |