On the distribution of scattering poles for perturbations of the Laplacian

Vodev, Georgi

Annales de l'Institut Fourier, Tome 42 (1992) no. 3, pp. 625-635.

On considère des opérateurs autoadjoints et positifs de la forme $-\Delta +P$ (sui ne sont pas nécessairement elliptiques) dans ${ℝ}^n$, $n\ge 3$, où $P$ est un opérateur différentiel du deuxième ordre, à coefficients à support compact. On montre que le nombre des pôles de la diffusion en dehors d’un voisinage conique de l’axe réel admet des estimations semblables au cas elliptique.

We consider selfadjoint positively definite operators of the form $-\Delta +P$ (not necessarily elliptic) in ${ℝ}^n$, $n\ge 3$, odd, where $P$ is a second-order differential operator with coefficients of compact supports. We show that the number of the scattering poles outside a conic neighbourhood of the real axis admits the same estimates as in the elliptic case. More precisely, if $\left\{{\lambda }_j\right\}(Im{\lambda }_j\ge 0_{)}$ are the scattering poles associated to the operator $-\Delta +P$ repeated according to multiplicity, it is proved that for any $\epsilon >0$ there exists a constant $C_{\epsilon }>0$ so that $\#\{{\lambda }_j:|{\lambda }_j|\le r$, $\epsilon \le arg{\lambda }_j\le \pi -\epsilon \}\le C_{\epsilon }{r}^n$ for any $r\ge 1$.

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