MCS-MP seminar: Jérémy Faupin

Speaker: Jérémy Faupin (Lorraine)

Title: On the number of bound states for the fractional Schrödinger operator with a super-critical exponent

Abstract: We will consider in this talk the number of negative eigenvalues $N_{<0}(H_s)$ of the fractional Schrödinger operator $H_s=(-\Delta)^s-V(x)$ in $L^2(\mathbb{R}^d)$, in any dimension $d\ge 1$ and for any $s>0$. After recalling results in the subcritical case $s<d/2$, including the celebrated Cwikel-Lieb-Rozenblum bounds for dimensions $d\ge3$ and Bargmann’s estimate in dimension $d=1$, we will focus on the supercritical case $s\ge d/2$. We will describe bounds on $N_{<0}(H_s)$ depending on $s-d/2$ being an integer or not, the critical case $s=d/2$ requiring a further analysis. This is joint work with Sébastien Breteaux and Viviana Grasselli.