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We study the well-known Friedrichs model, in which a discrete state is coupled to a continuum state.
K. O. Friedrichs, Commun. Pure Appl. Math., 1
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Z.-Y. Zhou and Z. Xiao, Phys. Rev., D 84
Cited in the paper.
For F ( z ) F(z) real on the negative axis and complex on the positive real axis, F ( z ) F(z) can be decomposed as F ( z ) = R ( z ) + i G ( z ) F(z)=R(z)+iG(z) , R ( z ) = Re F ( z ) R(z)={\rm Re}F(z) , G ( z ) = Im F ( z ) G(z)={\rm Im}F(z) , for z ∈ ℝ + z\in\mathbb{R}_{+} . R ( z ) R(z) and G ( z ) G(z) are the analytic continuations of Re F ( z ) {\rm Re}F(z) , Im F ( z ) {\rm Im}F(z) , respectively, from the positive axis. From Schwartz reflection principle and the uniqueness of the analytic continuation, F ∗ ( z ) = F ( z ∗ ) F^{*}(z)=F(z^{*}) , R ∗ ( z ) = R ( z ∗ ) R^{*}(z)=R(z^{*}) , G ∗ ( z ) = − G ( z ∗ ) G^{*}(z)=-G(z^{*}) on the first sheet. However, the Schwartz reflection principle also requires that the continuation of G ( z ) G(z) satisfies G ∗ ( z ) = G I I ( z ∗ ) G^{*}(z)=G^{II}(z^{*}) , which means G ( z ) = − G I I ( z ) G(z)=-G^{II}(z)
Cited in the paper.
2014
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Z.-Y. Zhou and Z. Xiao, Phys. Rev., D 83
2072
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