Upper bounds on the number of scattering poles and the Lax-Phillips conjecture. (English) Zbl 0801.35099

One considers a second order differential operator \(G\) in an exterior domain \(\Omega\subset \mathbb{R}^ n\). It is supposed that \(G\) is a perturbation of the operator \(-\Delta\) by a differential operator with compactly supported coefficients. Then the resolvent \((G- z^ 2)^{- 1}\), well defined for \(z\) from the lower half-plane, admits an analytic continuation on the half-plane \(\text{Im }z\geq 0\).
The goal of the paper is to find upper bounds on the number of poles \(z_ i\) of this operator function. It is shown that the counting function satisfies for any \(r\geq 1\) and \(\gamma>0\) the inequality \[ \#\{z_ j: | z_ j|\leq r,\;\text{Im }z_ j\geq \gamma\}\leq C\gamma^{-1} r^{n+1}+ C_ \gamma r^ n. \]


35P25 Scattering theory for PDEs
47F05 General theory of partial differential operators