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