On the number of bound states for fractional Schr{\"o}dinger operators with critical and super-critical exponent

TL;DR

This paper investigates the number of negative eigenvalues of fractional Schrödinger operators with critical and super-critical exponents, providing bounds that depend on the relation between the fractional order and the spatial dimension.

Contribution

It establishes bounds on the negative eigenvalues of fractional Schrödinger operators for all dimensions and fractional orders, including the critical case, using novel spectral analysis techniques.

Findings

Derived bounds on negative eigenvalues depending on s and d

Handled the critical case s=d/2 with a Cwikel-type estimate

Split the Birman-Schwinger operator into low- and high-energy parts

Abstract

We study the number $N_{<0}(H_s)$ of negative eigenvalues, counting multiplicities, of the fractional Schr\"odinger operator $H_s=(-\Delta)^s-V(x)$ on $L^2(\mathbb{R}^d)$, for any $d\ge1$ 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.