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              <p>We study the number N&amp;lt;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&amp;lt;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.</p>
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