Forward and backward problems for abstract time-fractional Schrödinger equations
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909823858114560 |
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| author | Chorfi, S. E. Et-tahri, F. Maniar, L. Yamamoto, M. |
| author_facet | Chorfi, S. E. Et-tahri, F. Maniar, L. Yamamoto, M. |
| contents | We investigate forward and backward problems associated with abstract time-fractional Schrödinger equations $\mathrm{i}^ν\partial_t^αu(t) + A u(t)=0$, $α\in (0,1)\cup (1,2)$ and $ν\in\{1,α\}$, where $A$ is a self-adjoint operator with compact resolvent on a Hilbert space $H$. This kind of equation, which incorporates the Caputo time-fractional derivative of order $α$, models quantum systems with memory effects and anomalous wave propagation. We first establish the well-posedness of the forward problems in two scenarios: ($ν=1,\,$ $α\in (0,1)$) and ($ν=α,\,$ $α\in (0,1)\cup (1,2)$). Then, we prove well-posedness and stability results for the backward problems depending on the two cases $ν=1$ and $ν=α$. Our approach employs the solution's eigenvector expansion along with the properties of the Mittag-Leffler functions, including the distribution of zeros and asymptotic expansions. Finally, we conclude with a discussion of some open problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_03600 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Forward and backward problems for abstract time-fractional Schrödinger equations Chorfi, S. E. Et-tahri, F. Maniar, L. Yamamoto, M. Analysis of PDEs Mathematical Physics 35R11, 35R30, 35R25 We investigate forward and backward problems associated with abstract time-fractional Schrödinger equations $\mathrm{i}^ν\partial_t^αu(t) + A u(t)=0$, $α\in (0,1)\cup (1,2)$ and $ν\in\{1,α\}$, where $A$ is a self-adjoint operator with compact resolvent on a Hilbert space $H$. This kind of equation, which incorporates the Caputo time-fractional derivative of order $α$, models quantum systems with memory effects and anomalous wave propagation. We first establish the well-posedness of the forward problems in two scenarios: ($ν=1,\,$ $α\in (0,1)$) and ($ν=α,\,$ $α\in (0,1)\cup (1,2)$). Then, we prove well-posedness and stability results for the backward problems depending on the two cases $ν=1$ and $ν=α$. Our approach employs the solution's eigenvector expansion along with the properties of the Mittag-Leffler functions, including the distribution of zeros and asymptotic expansions. Finally, we conclude with a discussion of some open problems. |
| title | Forward and backward problems for abstract time-fractional Schrödinger equations |
| topic | Analysis of PDEs Mathematical Physics 35R11, 35R30, 35R25 |
| url | https://arxiv.org/abs/2510.03600 |