The fractional logarithmic Schrödinger operator: properties and functional spaces
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909163724996608 |
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| author | Feulefack, Pierre Aime |
| author_facet | Feulefack, Pierre Aime |
| contents | In this note, we deal with the fractional Logarithmic Schrödinger operator $(I+(-Δ)^s)^{\log}$ and the corresponding energy spaces for variational study. The fractional (relativistic) Logarithmic Schrödinger operator is the pseudo-differential operator with logarithmic Fourier symbol, $\log(1+|ξ|^{2s})$, $s>0$. We first establish the integral representation corresponding to the operator and provide an asymptotics property of the related kernel. We introduce the functional analytic theory allowing to study the operator from a PDE point of view and the associated Dirichlet problems in an open set of $\mathbb{ R}^N.$ We also establish some variational inequalities, provide the fundamental solution and the asymptotics of the corresponding Green function at zero and at infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_02481 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The fractional logarithmic Schrödinger operator: properties and functional spaces Feulefack, Pierre Aime Analysis of PDEs 45P05, 45C05, 45A05, 35R11, 35S15 In this note, we deal with the fractional Logarithmic Schrödinger operator $(I+(-Δ)^s)^{\log}$ and the corresponding energy spaces for variational study. The fractional (relativistic) Logarithmic Schrödinger operator is the pseudo-differential operator with logarithmic Fourier symbol, $\log(1+|ξ|^{2s})$, $s>0$. We first establish the integral representation corresponding to the operator and provide an asymptotics property of the related kernel. We introduce the functional analytic theory allowing to study the operator from a PDE point of view and the associated Dirichlet problems in an open set of $\mathbb{ R}^N.$ We also establish some variational inequalities, provide the fundamental solution and the asymptotics of the corresponding Green function at zero and at infinity. |
| title | The fractional logarithmic Schrödinger operator: properties and functional spaces |
| topic | Analysis of PDEs 45P05, 45C05, 45A05, 35R11, 35S15 |
| url | https://arxiv.org/abs/2310.02481 |