A problem to account for the direct electron-positron annihilation in positron-hydrogen scattering above the positronium formation threshold has been resolved within the time independent formalism. The generalization of the optical theorem is derived for the case when an absorption potential is present in the Hamiltonian. With this theorem the annihilation cross section is fully determined by scattering amplitudes. This allows us to separate out analytically the contribution of the positronium formation from the overall annihilation cross section. The rest is determined as the direct annihilation cross section. It is done uniformly below as well as above the positronium formation threshold. The multichannel three-body theory for scattering states in the presence of an imaginary absorption potential is developed in order to compute the direct $e^+ e^-$ annihilation amplitude. Special attention has been paid to an accurate definition of the coordinate part of the absorption potential as the properly constructed zero-range potential, which corresponds to the delta-function originated from the first order perturbation theory. The calculated direct annihilation cross section below the positronium formation threshold is in good agreement with results of other authors. The direct annihilation cross section computed with the formalism of the paper shows nonsingular behavior at the positronium formation threshold and is in good agreement with existing data. A number of $e^+ e^-$ direct annihilation cross sections and positronium formation cross sections in the energy gap between Ps$(1s)$ and H$(n=2)$ thresholds are reported. A sharp increase in the calculated direct annihilation cross section across the resonant energy is found for all first S and P-wave Feshbach resonances.
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机译:在与时间无关的形式主义内,解决了在高于正电子形成阈值的正电子-氢散射中考虑直接电子-正电子account灭的问题。对于哈密顿量中存在吸收势的情况,可以推导光学定理的一般化。利用该定理,an灭截面完全由散射振幅确定。这使我们能够从整个an没截面上分析性地分离出the形成的贡献。其余的确定为直接an没的横截面。均匀地在below形成阈值之上和之下进行。为了计算直接的$ e ^ + e ^-$ an灭幅度,发展了一种在虚吸收势存在下散射状态的多通道三体理论。已经特别注意将吸收电势的坐标部分的准确定义定义为正确构造的零范围电势,该零范围电势对应于源自一阶扰动理论的增量函数。在正电子形成阈值以下计算的直接an没横截面与其他作者的结果非常吻合。用本文的形式主义计算的直接an没横截面显示出在正电子形成阈值处的非奇异行为,并且与现有数据非常吻合。报道了在Ps $(1s)$和H $(n = 2)$阈值之间的能隙中的许多$ e ^ + e ^-$直接an没横截面和形成横截面。对于所有第一个S波和P波Feshbach共振,发现在整个共振能中,计算出的直接hil没横截面急剧增加。
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