Summary
The problem of representing scattering data given on the boundary of the analyticity domain by analytic functions satisfying unitarity is investigated. The optimal representation inL 2-norm is shown to be the solution of a constrained convex optimization problem in some Hilbert space of analytic functions. A duality optimization theorem based on the generalized Lagrange multiplier technique is applied for solving this problem. The method is found to easily accommodate unequal errors on the real and imaginary parts of the amplitude and the case of data given along a limited part of the boundary.
Riassunto
Si prende in esame il problema di rappresentare per mezzo di funzioni analitiche che soddisfano l’unitarietà i dati di scattering ottenuti sul confine del dominio di analiticità. Si mostra come la rappresentazione ottimale nella normaL 2 sia la soluzione di un problema di ottimizzazione con condizioni di convessità in uno spazio di Hilbert di funzioni analitiche. Per risolvere questo problema si applica un teorema di ottimizzazione della dualità basato sulla tecnica dei moltiplicatori di Lagrange generalizzata. Si trova il metodo di armonizzare il caso di errori disuguali per le parti reale e immaginaria dell’ampiezza e il caso di dati riferentisi a una parte limitata del confine.
Реэюме
Исследуется проблема представления данных по рассеянию, эаданных на границе области аналитичности, с помошью аналитической функции, удовлетворяюшей унитарности. Покаэывается, что оптимальное представление вL 2-норме представляет рещение проблемы ограниченной выпуклой оптимиэации в том же гильбертовом пространстве аналитических функций. Теорема оптимиэации дуальности, основанная на обобшенной технике множителей Лагранжа, применяется для рещения зтой проблемы. Обнаружено, что зтот метод легко приспособить для неравных ощибок для вешественной и мнимой частей амплитуды и для случая, когда данные эаданы вдоль ограниченной части границы.
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Actually, the use of positivity constraints may by itself be of considerable help in making the extrapolations stable. Some examples of such extrapolations may be found in (8).
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Actually, when dealing with amplitudes at fixed positive momentum transfer (for instance the Legendre expansion inside the Lehman-Martin elipse), the unitarity condition (3b) automatically extends over the whole boundary. However, in order to use unitarity as a stabilizing condition in a more general context, one could consider the scattering amplitude in terms of a new complex variable (16) (i.e. a rational function (16) of the usual variabless, t, u) along a suitable curve drawn in the Mandelstam plane. For the purpose of analytical extrapolation the partΓ 2 of the curve along which data are missing should be forced to enter regions in the Mandelstam plane where the unitarity condition (3) applies. In this way stable analytic extrapolations to unphysical points using simultaneously information from crossed channels, as well as compatibility tests of data with crossing and unitarity are feasible.
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Caprini, I. On the best representation of scattering data by analytic functions inL 2-norm with positivity constraints. Nuov Cim A 21, 236–248 (1974). https://doi.org/10.1007/BF02724804
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DOI: https://doi.org/10.1007/BF02724804