Communications in Mathematical Physics

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

Analyticity properties of eigenfunctions and scattering matrix

  • Erik Balslev


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).


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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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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