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

We study the number N_<0(H_s) of negative eigenvalues, counting multiplicities, of the fractional Schrödinger operator H_s=(-Δ)^s-V(x) on L^2(ℝ^d), for any d≥1 and s≥ d/2. We prove a bound on N_<0(H_s) 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 ℒ^2,∞ to handle the high-energies part.