Summary
Hilbert-space versions of the Lippmann-Schwinger equations for wave operators and the usualT-matrix formulae are derived by using the theory of Riemann-Stieltjes integrals of operator-valued functions with respect to spectral functions. These results are valid for the renormalized wave operators, which have to be introduced when the interactions are of long range.
Riassunto
Si fa uso della teoria degli integrali di Riemann-Stieltjes di funzioni con valori di operatori rispetto alle funzioni spettrali per dedurre la versione nello spazio hilbertiano delle equazioni di Lippmann-Schwinger per gli operatori d’onda e le consuete formule della matriceT. Questi risultati sono validi per gli operatori d’onda rinormalizzati che devono essere introdotti quando l’interazione è a lungo raggio.
Резюме
Используя теорию интегралов Римана-С тильтьеса для операторных функций в отношении спектра льных функций, вывод ятся варианты уравнений Липпман а-Швингера в гильбер товом пространстве для операторов волны и формулы обычной Т-м атрипы. Эти результат ы справедливы для пе ренормированных вол-новых-операторов, которые должны быть введены, когда взаим одействия являются длиннодействующими.
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References
E. Prugovečki:Nuovo Cimento,63 B, 569 (1969).
E. Prugovečki:Riemann-Stieltjes integration of operator-valued functions with respect to vector-valued junctions on Banach spaces, SIAM Journ. Math. Anal. (1971-72) (to appear).
E. Prugovečki:Nuovo Cimento,4 B, 105 (1971).
E. Prugovečki:Quantum Mechanics in Hilbert Space (New York, 1971).
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Supported in part by the National Research Council of Canada.
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Prugovečki, E. Integral representations of the wave and transition operators in nonrelativistic scattering theory. Nuov Cim B 4, 124–134 (1971). https://doi.org/10.1007/BF02737570
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DOI: https://doi.org/10.1007/BF02737570