Summary
A thorough singularity analysis of scattering wave functions in theN-body problem is carried out. Operating on the basic Lippmann-Schwinger-Glöckle-Tobocman equations and resorting to a general reversion property for linked-cluster strings of channel resolvents and interactions, we explicitly exhibit the sources of the primary singularities ofN-body scattering wave functions. We introduce complete sets of eigenstates of the two-cluster channel Hamiltonians and we arrive at momentum-space representations for two-cluster wave function components, where all the primary singularities appear in separate terms. By means of simple algebraic procedures we show that appropriate sums of the residues at the physical poles lead to the physical transition amplitudes for elastic/rearrangement, partial and total break-up processes.
Riassunto
In questo lavoro si compie un'analisi completa e dettagliata delle singolarità presenti nelle funzioni d'onda di scattering per il problema aN corpi. Mediante opportune transformazioni delle equazioni di Lippmann-Schwinger-Glöckle-Tobocman, e ricorrendo a una proprietà generale d'inversione per operatori costituiti da risolventi e interazioni, sono messe in evidenza le sorgenti delle singolarità primarie delle funzioni d'onda di scattering aN corpi. Si utilizzano sistemi di base di autostati degli hamiltoniani di canale per partizioni a due «cluster» e si arriva ad opportune rappresentazioni nello spazio dei momenti delle componenti a due «cluster» delle funzioni d'onda, nelle quali tutte le singolarità primarie appaiono in termini distinti. Per mezzo di semplici procedimenti algebrici si mostra che le ampiezze fisiche di transizione (per processi elastici, di riarrangiamento e di frammentazione, sia parziale che totale) si possono ottenere sommando opportunamente i resiiui delle funzioni d'onda ai poli fisici.
Резюме
Проводится анализ сингулярност∈й волновых функций рассеяния в проблемеN-тел. На основе уравнений Линпманна-Швингера-Глёкля-Тобокмана и, используя общее свойство обращения для связанных кластерных струн резольвент и взаимодействий, мы в явном виде показываем источники первичных сингулярностей волновых функцийN-частичного рассеяния. Мы вводим полную систему собственных состояний для двухкластерных Гамильтонианов и приходим к импульсно-координатным представлениям для компонент двухкластерных волновых функций, где все первичные сингулрности появляются в отдельных членах. С помощью простых алгебраических процедур мы показываем, что соответстующие суммы вычетов в физических полюсах приводят к амплитудам физических переходов для упругой перегруппировки, парциальных и полных процессов раснпада.
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Partially supported by Ministero della Pubblica Istruzione, Italy.
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Cattapan, G., Vanzani, V. Singularity analysis ofN-body scattering wave functions. Nuov Cim A 51, 509–531 (1979). https://doi.org/10.1007/BF02776496
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DOI: https://doi.org/10.1007/BF02776496