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

Abstract

We study the scattering poles of $\sqrt{-\Delta }+V$, 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 $\Lambda$ of $\log z$ as an operator $L^2\to L^2$. We then obtain the upper bound on the counting function $N(r,a)=\#\{z_j\in \Lambda :0\le |z_j|\le r,|\arg z_j|\le a\}$, $r>1$, $|a|>1$ as $C⟨a⟩(⟨r{⟩}^d+(\log ⟨a⟩{)}^d)$, where $z_j$ are the poles of $\chi R_V\chi$.