Self-adjoint extensions of symmetric relations associated with systems of ordinary differential equations with distributional coefficients
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915788060884992 |
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| author | Redolfi, Steven Weikard, Rudi |
| author_facet | Redolfi, Steven Weikard, Rudi |
| contents | We study the extension theory for the two-dimensional first-order system $Ju' +qu = wf$ of differential equations on the real interval $(a,b)$ where $J$ is a constant, invertible, skew-hermitian matrix and $q$ and $w$ are matrices whose entries are real distributions of order $0$ with $q$ hermitian and $w$ non-negative. Specifically, we characterize the boundary conditions for solutions $u$ in the closure of the minimal relation, as well as describe the properties of quasi-boundary conditions which yield self-adjoint extensions. We then apply these ideas to a popular extension of non-negative minimal relations: the Krein-von Neumann extension. For more context on how the Krein-von Neumann is defined, an appendix shows a construction of the Friedrichs extension from which the Krein-von Neumann is traditionally defined. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09152 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Self-adjoint extensions of symmetric relations associated with systems of ordinary differential equations with distributional coefficients Redolfi, Steven Weikard, Rudi Spectral Theory Mathematical Physics Classical Analysis and ODEs 34L05, 47B25, 47A06 We study the extension theory for the two-dimensional first-order system $Ju' +qu = wf$ of differential equations on the real interval $(a,b)$ where $J$ is a constant, invertible, skew-hermitian matrix and $q$ and $w$ are matrices whose entries are real distributions of order $0$ with $q$ hermitian and $w$ non-negative. Specifically, we characterize the boundary conditions for solutions $u$ in the closure of the minimal relation, as well as describe the properties of quasi-boundary conditions which yield self-adjoint extensions. We then apply these ideas to a popular extension of non-negative minimal relations: the Krein-von Neumann extension. For more context on how the Krein-von Neumann is defined, an appendix shows a construction of the Friedrichs extension from which the Krein-von Neumann is traditionally defined. |
| title | Self-adjoint extensions of symmetric relations associated with systems of ordinary differential equations with distributional coefficients |
| topic | Spectral Theory Mathematical Physics Classical Analysis and ODEs 34L05, 47B25, 47A06 |
| url | https://arxiv.org/abs/2602.09152 |