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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.13397 |
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Table of 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.