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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2402.07675 |
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| _version_ | 1866916426670931968 |
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| author | Green, William R. Lane, Connor Lyons, Benjamin Ravishankar, Shyam Shaw, Aden |
| author_facet | Green, William R. Lane, Connor Lyons, Benjamin Ravishankar, Shyam Shaw, Aden |
| contents | We investigate dispersive estimates for the massless three dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies a $\langle t\rangle^{-1}$ decay rate as an operator from $L^1$ to $L^\infty$ regardless of the existence of zero energy eigenfunctions. We also show this decay rate may be improved to $\langle t\rangle ^{-1-γ}$ for any $0\leq γ< 1/2$ at the cost of spatial weights. This estimate, along with the $L^2$ conservation law allows one to deduce a family of Strichartz estimates in the case of a threshold eigenvalue. We classify the structure of threshold obstructions as being composed of zero energy eigenfunctions. Finally, we show the Dirac evolution is bounded for all time with minimal requirements on the decay of the potential and smoothness of initial data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07675 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Massless Dirac Equation in Three Dimensions: Dispersive estimates and zero energy obstructions Green, William R. Lane, Connor Lyons, Benjamin Ravishankar, Shyam Shaw, Aden Analysis of PDEs Mathematical Physics We investigate dispersive estimates for the massless three dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies a $\langle t\rangle^{-1}$ decay rate as an operator from $L^1$ to $L^\infty$ regardless of the existence of zero energy eigenfunctions. We also show this decay rate may be improved to $\langle t\rangle ^{-1-γ}$ for any $0\leq γ< 1/2$ at the cost of spatial weights. This estimate, along with the $L^2$ conservation law allows one to deduce a family of Strichartz estimates in the case of a threshold eigenvalue. We classify the structure of threshold obstructions as being composed of zero energy eigenfunctions. Finally, we show the Dirac evolution is bounded for all time with minimal requirements on the decay of the potential and smoothness of initial data. |
| title | The Massless Dirac Equation in Three Dimensions: Dispersive estimates and zero energy obstructions |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2402.07675 |