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

Abstract

We study the number $N_{<0}(H_s)$ of negative eigenvalues, counting multiplicities, of the fractional Schrödinger operator $H_s=(-\Delta {)}^s-V(x)$ on $L^2({\mathbb{R}}^d)$, for any $d\ge 1$ and $s\ge d/2$. 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 ${\mathcal{L}}^{2,\infty }$ to handle the high-energies part.