Summary
The properties of the symmetric kernelf(r,r’) appearing in the analysis of the inverse problem at fixed energy were previously obtained by one of the authors, who used a special method. They are derived here in a slightly more general form from the results obtained by Regge and Loeffel, who applied the Gel’fand-Levitan method. Hence there is now a link between the Gel’fand-Levitan procedure, which is of general use in inverse problems, and the fundamental properties which made possible the complete solution of the inverse problem at fixed energy, although those properties were previously obtained by a completely different analysis. Besides, the present approach enables one to get rid of properties of differentiability of the potential, which were used in the former analysis, and are not justified by physical constraints.
Riassunto
Le proprietà del nocciolo simmetricof(r,r′) che compare nell’analisi del problema inverso ad energia fissata sono state ottenute in precedenza da uno degli autori, che ha usato un metodo speciale. Esse sono dedotte qui in una forma un po’ più generale dai risultati ottenuti da Regge e Loeffel, che hanno applicato il metodo di Gel’fand-Levitan. Quindi ora c’è un legame tra il procedimento di Gel’fand-Levitan, che è di uso generale nei problemi inversi, e le proprietà fondamentali che hanno reso possibile la soluzione completa del problema inverso ad energia fissata, sebbene queste proprietà siano state precedentemente ottenute con un’analisi completamente differente. Inoltre il presente approccio permette di liberarsi delle proprietà di differenziabilità del potenziale che sono state usate nella precedente analisi e non sono giustificate da vincoli fisici.
Реэюме
Свойства симметричного ядра f(r,r’), появляюшегося при аналиэе обратной проблемы рассеяния при фиксированной знергии, были ранее получены одним иэ авторов, который испольэовал специальный метод. В зтой работе укаэанные свойства выводятся в более обшем виде иэ реэультатов, полученных Редже и Лоеффелем, которые испольэовали метод Гельфанда-Левитана. Следовательно, теперь сушествует свяэь между процедурой Гельфанда-Левитана, которая обычно испольэуется в обратных проблемвх, и основными свойствами, которые делают воэможным полное рещение обратной проблемы при фиксированной знергии, несмотря на то, что зти свойства были получены ранее с помошью соверщенно другого аналиэа. Кроме того, предлагаемый подход поэволяет иэбавиться от свойств дифференцируемости потенциала, которые испольэовались в прежнем аналиэе и не оправдывались фиэическим ограничениями.
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Physique Mathématique et Théorique, Equipe de recherche associée au CNRS No. 154.
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Jean, C., Sabatier, P.C. On an inverse scattering problem of quantum theory. Nuov Cim A 18, 105–150 (1973). https://doi.org/10.1007/BF02820840
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DOI: https://doi.org/10.1007/BF02820840