Summary
A rigorous analysis of the foundations and of the properties of the theoretical optical potential is attempted. It is shown that, under suitable conditions on the two-body potentials, the optical potential exists both in a time-dependent and in a time-independent theory and is a bounded operator. This property does not require the boundedness of the two-body potentials. The analytic and asymptotic properties of the optical potential in the energy plane are obtained. The connection between the time-independent and time-dependent formulation is discussed in a rigorous way.
Riassunto
In questo lavoro si effettua un tentativo di analisi rigorosa dei fondamenti e delle proprietà del potenziale ottico. Si dimostra che, se i potenziali a due corpi soddisfano ad opportune condizioni, allora il potenziale ottico esiste ed è un operatore limitato sia nella teoria dipendente dal tempo che in quella indipendente dal tempo. Questa proprietà non richiede che i potenziali a due corpi siano limitati. Si ottengono inoltre le proprietà di analiticità e il comportamento asintotico del potenziale ottico nel piano dell’energia. Infine si discute in modo rigoroso la connessione tra la formulazione dipendente dal tempo e quella indipendente dal tempo.
Реэюме
Проводится строгий аналиэ обоснований и свойств теоретического оптического потенциала. Покаэывается, что при соответствуюших условиях на двух-частичные потенциалы, оптический потенциал сушествует и в теории, не эависяшей от времени, и в теории, эависяшей от времени, и представляет ограниченный оператор. Это свойство не требует ограниченности двух-частичных потенциалов. Выводятся аналитические и асимптотические свойства оптического потенциала в плоскости знергии. Аккуратно обсуждается свяэь между формулировкой, не эависяшей от времени, и формулировкой, эависяшей от времени.
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This relation can easily be proved by means of the spectral resolution of the resolvent of a self-adjoint operator — see, for instance (10), p. 324.
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Bertero, M., Passatore, G. A mathematical approach to the derivation of the theoretical optical potential. Nuov Cim A 2, 579–604 (1971). https://doi.org/10.1007/BF02736736
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DOI: https://doi.org/10.1007/BF02736736