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Analyticity properties of eigenfunctions and scattering matrix. (English) Zbl 0662.35079
For potentials $$V=V(x)=O(| x|^{-2-\epsilon})$$ for $$| x| \to \infty$$, $$x\in {\mathbb{R}}^ 3$$, we prove that if the S-matrix of $$(- \Delta,-\Delta +V)$$ has an analytic extension $$\tilde S(z)$$ to a region $${\mathcal O}$$ in the lower half-plane, then the family of generalized eigenfunctions of $$-\Delta +V$$ has an analytic extension $${\tilde \phi}$$(k,$$\omega$$,x) to $${\mathcal O}$$ such that $$| {\tilde \phi}(k,\omega,x)| <Ce^{b| x|}$$ for $$| Im k| <b$$. Consequently, the resolvent $$(-\Delta +V-z^ 2)^{-1}$$ has an analytic continuation from $${\mathbb{C}}^+$$ to $$\{$$ $$k\in {\mathcal O}|$$ $$| Im k| <b\}$$ as an operator $$\tilde R(z)$$ from $${\mathcal H}_ b=\{f=e^{- b| x|}g | \quad g\in L_ 2({\mathbb{R}}^ 3)\}$$ to $${\mathcal H}_{-b}$$. Based on this, we define for potentials $$W=o(e^{-2b| x|})$$ resonances of $$(-\Delta +V,-\Delta +V+W)$$ as poles of $$(1+W\tilde R(z))^{-1}$$ and identify these resonances with poles of the analytically continued S-matrix of $$(-\Delta +V,-\Delta +V+W)$$.

##### MSC:
 35P25 Scattering theory for PDEs 35A20 Analyticity in context of PDEs 35Q99 Partial differential equations of mathematical physics and other areas of application 81U20 $$S$$-matrix theory, etc. in quantum theory
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