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Pré-Publication, Document De Travail Année : 2024

On the number of bound states for fractional Schrödinger operators with critical and super-critical exponent

Jérémy Faupin
Viviana Grasselli
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Résumé

We study the number N<0 (Hs ) of negative eigenvalues, counting multiplicities, of the fractional Schrödinger operator Hs = (−∆)^s − V (x) on L2 (Rd), for any d ≥ 1 and s ≥ d/2. We prove a bound on N<0 (Hs ) which depends on s − d/2 being either an integer or not, the critical case s = d/2 requiring a further analysis. Our proof relies on a splitting of the Birman-Schwinger operator associated to this spectral problem into low- and high- energies parts, a projection of the low-energies part onto a suitable subspace, and, in the critical case s = d/2, a Cwikel-type estimate in the weak trace ideal L2,∞ to handle the high-energies part.
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Dates et versions

hal-04522608 , version 1 (26-03-2024)

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  • HAL Id : hal-04522608 , version 1

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Sébastien Breteaux, Jérémy Faupin, Viviana Grasselli. On the number of bound states for fractional Schrödinger operators with critical and super-critical exponent. 2024. ⟨hal-04522608⟩
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