On the number of bound states for fractional Schr{ö}dinger 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 $H_s=(-\mathrm{\Delta }{)}^s-V(x)$ on $L^2({\mathbb{R}}^d)$, for any $d\ge 1$ and $s\ge d\mathrm{/}2$. We prove a bound on $N_{<0}(H_s)$ which depends on $s-d\mathrm{/}2$ being either an integer or not, the critical case $s=d\mathrm{/}2$ 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 $s=d\mathrm{/}2$, a Cwikel-type estimate in the weak trace ideal ${\mathcal{L}}^{2,\mathrm{\infty }}$ to handle the high-energies part.