Summary
The occurrence of ε limits in the Lippmann-Schwinger integral equations of scattering have given rise to some common misconceptions. The two- and three-body problems are re-examined in some detail and it is shown that for the two-body problem an infinite sequence of integral equations with finite ε values may be replaced by a single equation in the limit of ε tending to zero. In the three-body problem an infinite sequence of integral equations can be replaced in the limit of ε tending to zero by a single inhomogenous equation together with two homogenous equations which need to be solved simultaneously. It is shown, furthermore, that these three equations are mutually compatible.
Riassunto
La presenza di limite per ε tendente a zero nelle equazioni integrali di Lippmann e Schwinger della teoria dello scattering ha dato luogo a qualche errore comune. I problemi a due e a tre corpi sono riesaminati in dettaglio e si dimostra che, per il problema a due corpi, una successione infinita di equazioni integrali con valori finiti di ε può essere sostituita da una singola equazione nel limite di ε tendente a zero. Nel problema a tre corpi, una successione infinita di equazioni integrali può essere sostituita nel limite di ε tendente a zero da una singola equazione non omogenea insieme con due equazioni omogenee che devono essere risolte simultaneamente. Si dimostra inoltre che queste tre equazioni sono mutuamente compatibili.
Резюме
Наличие ε-пределов в интегральных уравнениях Липпманна-Швингера для рассеянной волны приводит к некоторым обычным недоразумениям. заново подробно исследуйтся проблемы дв↑х и трех тел. Показывается, что для провлемы двух тел бесконечная последовательность интегральных уравнений с конечными значениями ε может быть заменена одним уравнением в пределе, когда ε стремится к нулю. В проблеме трех тел бесконечнал последовательность интегральных может быть заменена в пределе, когда ε стремится к нулю, одним неоднородным уравнением и двумя однородными уравнениями, которые следует решать одновременно. Затем показывается, что эти три уравнения являются совместимыми.
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Lovitch, L. On the Lippmann-Schwinger equations. Nuov Cim A 68, 81–100 (1982). https://doi.org/10.1007/BF02902634
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DOI: https://doi.org/10.1007/BF02902634