Summary
Let us consider the class of spherically symmetric potentials which are so smooth that the functionsf(k) are analytic in the lower half plane. The Gel’fand-Levitan equation is used to construct all potentials of this class which, for a given angular momentum, have scattering phases which are identically zero for all energy. In these cases the potentials support point eigenvalues which are either zero or are positive and embedded in the continuous spectrum. In the casel=0, the potential which supports the zero point eigenvalue was given earlier by Bargmann who used a special technique to find it.
Riassunto
Consideriamo la classe di quei potenziali a simmetria sferica che si raccordino così dolcemente che le funzionif(k) siano analitiche nel semipiano inferiore. Si usa l’equazione di Gel’fand-Levitan per costruire tutti i potenziali di questa classe che, per un dato momento angolare, hanno fasi di scattering identicamente nulle per ogni energia. In questi casi i potenziali sostengono autovalori puntuali o nulli o positivi inclusi nello spettro continuo. Nel casol=0 il potenziale che sostiene l’autovalore nullo fu dato precedentemente da Bargmann, che per trovarlo ricorse a una tecnica speciale.
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References
L. M. Gel’fand andB. M. Levitan:Isv. Akad. Nauk SSSR Math. Series,15, 309 (1951).
R. Jost andW. Kohn:Det. Kgl. Dan. Vidensk. Selsk., Mat.-fys. Medd.,27, no. 9 (1953).
N. Levinson:Det. Kgl. Dan. Vidensk. Selsk., Mat.-fys. Medd.,25, no. 9 (1949).
I. Kay andH. E. Moses:Nuovo Cimento,2, 917 (1955);3, 66 (1955).
I. Kay andH. E. Moses:Journ. App. Phys.,27, 1503 (1956).
V. Bargmann:Rev. Mod. Phys.,21, 488 (1949).
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The work of H.E.M. was sponsored by the Geophysics Research Directorate, Air Force Cambridge Research Center, Air Research and Development Command under Contract no. 926 AF 19(604) 4555.
The work of S.F.T. was supported in part by the Office of Naval Research and by the National Science Foundation (69520-12-20426).
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Moses, H.E., Tuan, S.F. Potentials with zero scattering phase. Nuovo Cim 13, 197–206 (1959). https://doi.org/10.1007/BF02727541
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DOI: https://doi.org/10.1007/BF02727541