On Fourier expansions for systems of ordinary differential equations with distributional coefficients
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866914316111839232 |
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| author | Redolfi, Steven Weikard, Rudi |
| author_facet | Redolfi, Steven Weikard, Rudi |
| contents | We study the spectral theory for the 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 distributions of order $0$ with $q$ hermitian and $w$ non-negative. Specifically, we construct a generalized Weyl-Titchmarsh $m$-function with corresponding spectral measure $τ$ and a generalized Fourier transform after imposing certain conditions on $J$, $q$, and $w$. Different conditions are motivated and studied in the later sections. A Fatou-type identity needed for our result is recorded in the appendix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_02086 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Fourier expansions for systems of ordinary differential equations with distributional coefficients Redolfi, Steven Weikard, Rudi Spectral Theory Mathematical Physics 34L10, 47A70 We study the spectral theory for the 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 distributions of order $0$ with $q$ hermitian and $w$ non-negative. Specifically, we construct a generalized Weyl-Titchmarsh $m$-function with corresponding spectral measure $τ$ and a generalized Fourier transform after imposing certain conditions on $J$, $q$, and $w$. Different conditions are motivated and studied in the later sections. A Fatou-type identity needed for our result is recorded in the appendix. |
| title | On Fourier expansions for systems of ordinary differential equations with distributional coefficients |
| topic | Spectral Theory Mathematical Physics 34L10, 47A70 |
| url | https://arxiv.org/abs/2303.02086 |