On the definition of zero resonances for the Schr{ö}dinger operator with optimal scaling potentials
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917618415304704 |
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| author | Grasselli, Viviana |
| author_facet | Grasselli, Viviana |
| contents | We consider the Schr{ö}dinger operator --$Δ$ + V on the Euclidean space with potential in the Lorentz space L^{n/2,1} and we find necessary and sufficient conditions for zero to be a resonance or an eigenvalue. We consider functions with gradient in L^2 and that verify the equation (--$Δ$ + V)$ψ$ = 0, namely the kernel of (--$Δ$ + V) in the homogeneous Sobolev space of order one. We prove that a function in this set is either in a weak Lebesgue space or in L^2 , in the latter case we have a zero eigenfunction. The set of eigenfunctions is the hyperplane of functions that are orthogonal to V, furthermore we show that under some classic orthogonality conditions a zero eigenfunction belongs to the weak Lebesgue space of order one or to L^1. We study dimensions n $\ge$ 3 and in dimension three we generalize a result proved by Beceanu. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13397 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the definition of zero resonances for the Schr{ö}dinger operator with optimal scaling potentials Grasselli, Viviana Spectral Theory We consider the Schr{ö}dinger operator --$Δ$ + V on the Euclidean space with potential in the Lorentz space L^{n/2,1} and we find necessary and sufficient conditions for zero to be a resonance or an eigenvalue. We consider functions with gradient in L^2 and that verify the equation (--$Δ$ + V)$ψ$ = 0, namely the kernel of (--$Δ$ + V) in the homogeneous Sobolev space of order one. We prove that a function in this set is either in a weak Lebesgue space or in L^2 , in the latter case we have a zero eigenfunction. The set of eigenfunctions is the hyperplane of functions that are orthogonal to V, furthermore we show that under some classic orthogonality conditions a zero eigenfunction belongs to the weak Lebesgue space of order one or to L^1. We study dimensions n $\ge$ 3 and in dimension three we generalize a result proved by Beceanu. |
| title | On the definition of zero resonances for the Schr{ö}dinger operator with optimal scaling potentials |
| topic | Spectral Theory |
| url | https://arxiv.org/abs/2403.13397 |