Article Dans Une Revue Osaka Journal of Mathematics Année : 2025

On the definition of zero resonances for the Schrödinger operator with optimal scaling potentials

Résumé

We consider the Schrödinger operator −∆ + 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 (−∆ + V)ψ = 0, namely the kernel of (−∆ + 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 ≥ 3 and in dimension three we generalize a result proved by Beceanu.

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Dates et versions

hal-04265475 , version 1 (30-10-2023)
hal-04265475 , version 2 (13-01-2024)
hal-04265475 , version 3 (20-02-2024)
hal-04265475 , version 4 (19-03-2024)

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Viviana Grasselli. On the definition of zero resonances for the Schrödinger operator with optimal scaling potentials. Osaka Journal of Mathematics, 2025, 62 (1), pp. 123-144. ⟨10.18910/100447⟩. ⟨hal-04265475v4⟩
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