Time-fractional discrete diffusion equation for Schrödinger operator
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910532844388352 |
|---|---|
| author | Dasgupta, Aparajita Mondal, Shyam Swarup Ruzhansky, Michael Tushir, Abhilash |
| author_facet | Dasgupta, Aparajita Mondal, Shyam Swarup Ruzhansky, Michael Tushir, Abhilash |
| contents | This article aims to investigate the semi-classical analog of the general Caputo-type diffusion equation with time-dependent diffusion coefficient associated with the discrete Schrödinger operator, $\mathcal{H}_{\hbar,V}:=-\hbar^{-2}\mathcal{L}_{\hbar}+V$ on the lattice $\hbar\mathbb{Z}^{n},$ where $V$ is a non-negative multiplication operator and $\mathcal{L}_{\hbar}$ is the discrete Laplacian. We establish the well-posedness of the Cauchy problem for the general Caputo-type diffusion equation with a regular coefficient in the associated Sobolev-type spaces. However, it is very weakly well-posed when the diffusion coefficient has a distributional singularity. Finally, we recapture the classical solution (resp. very weak) for the general Caputo-type diffusion equation in the semi-classical limit $\hbar\to 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13690 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Time-fractional discrete diffusion equation for Schrödinger operator Dasgupta, Aparajita Mondal, Shyam Swarup Ruzhansky, Michael Tushir, Abhilash Analysis of PDEs This article aims to investigate the semi-classical analog of the general Caputo-type diffusion equation with time-dependent diffusion coefficient associated with the discrete Schrödinger operator, $\mathcal{H}_{\hbar,V}:=-\hbar^{-2}\mathcal{L}_{\hbar}+V$ on the lattice $\hbar\mathbb{Z}^{n},$ where $V$ is a non-negative multiplication operator and $\mathcal{L}_{\hbar}$ is the discrete Laplacian. We establish the well-posedness of the Cauchy problem for the general Caputo-type diffusion equation with a regular coefficient in the associated Sobolev-type spaces. However, it is very weakly well-posed when the diffusion coefficient has a distributional singularity. Finally, we recapture the classical solution (resp. very weak) for the general Caputo-type diffusion equation in the semi-classical limit $\hbar\to 0$. |
| title | Time-fractional discrete diffusion equation for Schrödinger operator |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2402.13690 |