## 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 $\#\{z_ j: | z_ j|\leq r,\;\text{Im }z_ j\geq \gamma\}\leq C\gamma^{-1} r^{n+1}+ C_ \gamma r^ n.$

### MSC:

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