Effective upper bounds on the number of resonance in potential scattering
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866916489481682944 |
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| author | Cuenin, Jean-Claude |
| author_facet | Cuenin, Jean-Claude |
| contents | We prove upper bounds on the number of resonances and eigenvalues of Schrödinger operators $-Δ+V$ with complex-valued potentials, where $d\geq 3$ is odd. The novel feature of our upper bounds is that they are \emph{effective}, in the sense that they only depend on an exponentially weighted norm of V. Our main focus is on potentials in the Lorentz space $L^{(d+1)/2,1/2}$, but we also obtain new results for compactly supported or pointwise decaying potentials. The main technical innovation, possibly of independent interest, are singular value estimates for Fourier-extension type operators. The obtained upper bounds not only recover several known results in a unified way, they also provide new bounds for potentials which are not amenable to previous methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_06079 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Effective upper bounds on the number of resonance in potential scattering Cuenin, Jean-Claude Spectral Theory Mathematical Physics Analysis of PDEs Functional Analysis We prove upper bounds on the number of resonances and eigenvalues of Schrödinger operators $-Δ+V$ with complex-valued potentials, where $d\geq 3$ is odd. The novel feature of our upper bounds is that they are \emph{effective}, in the sense that they only depend on an exponentially weighted norm of V. Our main focus is on potentials in the Lorentz space $L^{(d+1)/2,1/2}$, but we also obtain new results for compactly supported or pointwise decaying potentials. The main technical innovation, possibly of independent interest, are singular value estimates for Fourier-extension type operators. The obtained upper bounds not only recover several known results in a unified way, they also provide new bounds for potentials which are not amenable to previous methods. |
| title | Effective upper bounds on the number of resonance in potential scattering |
| topic | Spectral Theory Mathematical Physics Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2209.06079 |