Bounds on the number of scattering poles of half-Laplacian in odd dimensions, $d\ge 3$

We study the scattering poles of $\sqrt{-\mathrm{\Delta }}+V$, where $V$ is a compactly supported, bounded and complex valued potential. We show that the resolvent operator $\chi R_V\chi$ has a meromorphic continuation to the whole Riemannian surface of $\mathrm{\Lambda }$ of $\log z$ as an operator $L^2\to L^2$. We then obtain the upper bound on the counting function $N(r,a)=\mathrm{\#}\{z_j\in \mathrm{\Lambda }:0\le \mathrm{\mid }{z}_j\mathrm{\mid }\le r,\mathrm{\mid }\arg z_j\mathrm{\mid }\le a\}$, $r>1$, $\mathrm{\mid }a\mathrm{\mid }>1$ as $C⟨a⟩(⟨r{⟩}^d+(\log ⟨a⟩{)}^d)$, where $z_j$ are the poles of $\chi R_V\chi$.