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 \((-\Delta +V,-\Delta +V+W)\).


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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