Modified Green-Hyperbolic Operators
Green-hyperbolic operators – partial differential operators on globally hyperbolic spacetimes that (together with their formal duals) possess advanced and retarded Green operators – play an important role in many areas of mathematical physics. Here, we study modifications of Green-hyperbolic operato...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2023 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2023
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211972 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Modified Green-Hyperbolic Operators. Christopher J. Fewster. SIGMA 19 (2023), 057, 27 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862633409277329408 |
|---|---|
| author | Fewster, Christopher J. |
| author_facet | Fewster, Christopher J. |
| citation_txt | Modified Green-Hyperbolic Operators. Christopher J. Fewster. SIGMA 19 (2023), 057, 27 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Green-hyperbolic operators – partial differential operators on globally hyperbolic spacetimes that (together with their formal duals) possess advanced and retarded Green operators – play an important role in many areas of mathematical physics. Here, we study modifications of Green-hyperbolic operators by the addition of a possibly nonlocal operator acting within a compact subset of spacetime, and seek corresponding '-nonlocal' generalised Green operators. Assuming the modification depends holomorphically on a parameter, conditions are given under which -nonlocal Green operators exist for all parameter values, with the possible exception of a discrete set. The exceptional points occur precisely where the modified operator admits nontrivial smooth homogeneous solutions that have past- or future-compact support. Fredholm theory is used to relate the dimensions of these spaces to those corresponding to the formal dual operator, switching the roles of future and past. The -nonlocal Green operators are shown to depend holomorphically on the parameter in the topology of bounded convergence on maps between suitable Sobolev spaces, or between suitable spaces of smooth functions. An application to the LU factorisation of systems of equations is described.
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| first_indexed | 2026-03-14T21:16:37Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211972 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T21:16:37Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Fewster, Christopher J. 2026-01-20T16:12:03Z 2023 Modified Green-Hyperbolic Operators. Christopher J. Fewster. SIGMA 19 (2023), 057, 27 pages 1815-0659 2020 Mathematics Subject Classification: 35R01; 35R09; 47B93 arXiv:2303.02993 https://nasplib.isofts.kiev.ua/handle/123456789/211972 https://doi.org/10.3842/SIGMA.2023.057 Green-hyperbolic operators – partial differential operators on globally hyperbolic spacetimes that (together with their formal duals) possess advanced and retarded Green operators – play an important role in many areas of mathematical physics. Here, we study modifications of Green-hyperbolic operators by the addition of a possibly nonlocal operator acting within a compact subset of spacetime, and seek corresponding '-nonlocal' generalised Green operators. Assuming the modification depends holomorphically on a parameter, conditions are given under which -nonlocal Green operators exist for all parameter values, with the possible exception of a discrete set. The exceptional points occur precisely where the modified operator admits nontrivial smooth homogeneous solutions that have past- or future-compact support. Fredholm theory is used to relate the dimensions of these spaces to those corresponding to the formal dual operator, switching the roles of future and past. The -nonlocal Green operators are shown to depend holomorphically on the parameter in the topology of bounded convergence on maps between suitable Sobolev spaces, or between suitable spaces of smooth functions. An application to the LU factorisation of systems of equations is described. It is a pleasure to thank Rainer Verch for useful discussions at various stages of this work and Maximilian Ruep for a careful reading and comments on a draft of the manuscript. I would like to particularly thank Christian Bär for posing the question that prompted Theorem 3.5, as well as Lashi Bandara and other participants in the conference "Global analysis on Manifolds" held in Bär’s honor (Freiburg, September 2022) for their useful remarks and conversations. I also thank the referees for their valuable suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Modified Green-Hyperbolic Operators Article published earlier |
| spellingShingle | Modified Green-Hyperbolic Operators Fewster, Christopher J. |
| title | Modified Green-Hyperbolic Operators |
| title_full | Modified Green-Hyperbolic Operators |
| title_fullStr | Modified Green-Hyperbolic Operators |
| title_full_unstemmed | Modified Green-Hyperbolic Operators |
| title_short | Modified Green-Hyperbolic Operators |
| title_sort | modified green-hyperbolic operators |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211972 |
| work_keys_str_mv | AT fewsterchristopherj modifiedgreenhyperbolicoperators |