Summary
The Bethe-Salpeter normalization condition is investigated and a unified picture of the different methods so far used is given together with their equivalence proof. The solutions of the Bethe-Salpeter equation are then examined in the Wick-Cutkosky model forn=l+1 for a check of the recent result of Nakanishi that abnormal solutions with oddx correspond to states with negative norm. This result is confirmed forn=l+1 and arbitraryx in the energy range 0<s≪n/(n+x)n+x+1) for which it is proved that the solution can be well approximated by the zero energy solution.
Riassunto
Si studia la condizione di normalizzazione dell'equazione di Bethe-Salpeter ed un quadro completo dei diversi metodi usati finora viene presentato insieme con la dimostrazione che tutti questi metodi sono equivalenti. Le soluzioni dell'equazione di Bethe-Salpeter sono considerate nell'ambito del modello di Wick-Cutkosky pern=l+1 allo scopo di verificare il risultato recentemente ottenuto da Nakanishi che le soluzioni anormali conx dispari corrispondono a stati di norma negativa. Questo risultato è provato perx qualsiasi nel dominio di energies≪n/(n+x)n+x+1) per il quale si dimostra che la soluzione è ben approssimata dalla soluzione corrispondente ad energia nulla.
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Literatur
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A preprint by the present author has been circulating (EFINS-65-35) which contained a mistake (Sect. 4) invalidating all the conclusions. The present paper is a revised version of the above-mentioned one.
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Generally speaking a solution of (3.4) will not be regular in more than one single singular point. We call eigenfunctions the solutions which are regular in at least two singular points. Mathematically this amounts to imposing the vanishing of a Wronskian and, therefore, to an eigenvalue problem.
F. G. Tricomi:Equazioni Differenziali (Torino, 1953). In the following we shall largely quote from this book and we shall refer to it as T followed by the appropriate page number.
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Ref. (13),F. G. Tricomi:Equazioni Differenziali (Torino, 1953). In the following we shall largely quote from this book and we shall refer to it as T followed by the appropriate page number., Sect.29. See alsoA. Bottino, A. M. Longoni andT. Regge:Nuoro Cimento,23, 954 (1962), Appendix I.
Ref. (13)F. G. Tricomi:Equazioni Differenziali (Torino, 1953). In the following we shall largely quote from this book and we shall refer to it as T followed by the appropriate page number, p. 57 and p. 77.
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Work supported by the Atomic Energy Commission under Contract No. AT (11-1)-264.
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Predazzi, E. Some remarks on the Bethe-Salpeter normalization properties. Nuovo Cimento A (1965-1970) 40, 913–927 (1965). https://doi.org/10.1007/BF02855994
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DOI: https://doi.org/10.1007/BF02855994