Reflectionless operators and automorphic Herglotz functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909428425424896 |
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| author | Remling, Christian |
| author_facet | Remling, Christian |
| contents | I am interested in canonical systems and Dirac operators that are reflectionless on an open set. In this situation, the half line $m$ functions are holomorphic continuations of each other and may be combined into a single function. By passing to the universal cover of its domain, we then obtain a one-to-one correspondence of these operators with Herglotz functions that are automorphic with respect to the Fuchsian group of covering transformations. I investigate the properties of this formalism, with particular emphasis given to the measures that are automorphic in a corresponding sense. This will shed light on the reflectionless operators as a topological space, on their extreme points, and on how the heavily studied smaller space of finite gap operators sits inside the (much) larger space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_10992 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reflectionless operators and automorphic Herglotz functions Remling, Christian Spectral Theory Mathematical Physics 34L40 81Q10 (primary), 30F35 (secondary) I am interested in canonical systems and Dirac operators that are reflectionless on an open set. In this situation, the half line $m$ functions are holomorphic continuations of each other and may be combined into a single function. By passing to the universal cover of its domain, we then obtain a one-to-one correspondence of these operators with Herglotz functions that are automorphic with respect to the Fuchsian group of covering transformations. I investigate the properties of this formalism, with particular emphasis given to the measures that are automorphic in a corresponding sense. This will shed light on the reflectionless operators as a topological space, on their extreme points, and on how the heavily studied smaller space of finite gap operators sits inside the (much) larger space. |
| title | Reflectionless operators and automorphic Herglotz functions |
| topic | Spectral Theory Mathematical Physics 34L40 81Q10 (primary), 30F35 (secondary) |
| url | https://arxiv.org/abs/2412.10992 |