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Communications in Mathematical Physics

, Volume 114, Issue 4, pp 599–612 | Cite as

Analyticity properties of eigenfunctions and scattering matrix

  • Erik Balslev
Article

Abstract

For potentialsV=V(x)=O(|x|−2−ε) for |x|→∞,x∈ℝ3 we prove that if theS-matrix of (−Δ, −Δ+V) has an analytic extension\(\tilde S(z)\) to a regionO in the lower half-plane, then the family of generalized eigenfunctions of −Δ+V has an analytic extension\(\tilde \phi (k,\omega ,x)\) toO such that\(\left| {\tilde \phi (k,\omega ,x)} \right|< Ce^{b\left| x \right|}\) for |Imk|<b. Consequently, the resolvent (−Δ+Vz2)−1 has an analytic continuation from ℂ+ to {kO‖Imk|<b} as an operator\(\tilde R(z)\) from ℋ b ={f=eb|x|g|gL2(ℝ3)} to ℋb. Based on this, we define for potentialsW=o(e−2b|x|) resonances of (−Δ+V, −Δ+V+W) as poles of\((1 + W\tilde R(z))^{ - 1}\) and identify these resonances with poles of the analytically continuedS-matrix of (−Δ+V, −Δ+V+W).

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Erik Balslev
    • 1
  1. 1.Denmark and Institute for Advanced StudyUniversity of AarhusPrincetonUSA

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