Summary
In this paper, using recurrence relations for the spherical Bessel functions, we deduce recurrence relations for the higher-order phase shifts\(\delta _\iota \) of three wide classes of potentials:U(r)=±r −m·f(r), wheref(r)>M>0 for allr,M being any fixed number;U(r)=±f(r)··exp [-cr 2], for a wide class of functionsf(r) described in the text, including potentials liker ±m·exp [−cr 2] and finite-range potentials;U(r)=±r ±m·f(r), 0<P<f(r)<Q, P andQ being any fixed numbers, forr≤a,U(r)=0 forr>a. The consequences of these relations are then briefly examined, one of these being that for these three classes of potentials\(\delta _{\iota - 1} /\delta _\iota \to I\), → μl and →μl 2, respectively, asl→∞.
Riassunto
In questo lavoro, usando le relazioni di ricorrenza per le funzioni sferiche di Bessel, si deducono le relazioni di ricorrenza per spostamenti di fase di ordine più alto,\(\delta _\iota \), di tre grandi classi di potenziali:U(r)=±r −m·f(r), dovef(r)>M>0 per tutti glir,M essendo qualsiasi numero fissato;U(r)=±f(r)·exp [-cr 2] per un'ampia classe di funzionif(r) descritte nel testo, che comprendono potenziali comer ±m·exp [−cr 2] e potenziali a raggio finito;U(r)=±r ±m·f(r), 0<P<f(r)<Q,P eQ essendo qualsiasi numero fissato, perr≤a,U(r=0 perr≥a. Si esaminano quindi in breve le conseguenze di queste relazioni, una di queste essendo che per queste tre classi di potenziali\(\delta _{\iota - 1} /\delta _\iota \to I\), rispettivamente, per l»∞.
Резуме
В этой работе, используя рекуррентные соотношения для сферических Бесселевых функций, мы выводим рекуррентные соотношения для фазовых сдвигов высших порядков\(\delta _\iota \) для трех классов потенциалов:U(r)=±r −m·f(r), гдеf(r)>M>0 для всехr, причемM любое фиксированное число,U(r)=±f(r)·exp [-cr 2], для широкого класса фуккцийf(x), описанных в тексте, причем эти классы включают потенциалы, подобныеr ±m·exp [-cr 2] и потенциалы с конечным радиусом,U(r)=±r ±m·f(r), 0<P<f(r)<Q, гдеP иQ являются фиксированными числами, дляr≤a,U(r)=0 дляr>a. Затем анализируются следствия зтих соотношений, одно из которых состоит в том, что для этих трех классов потенциалов соответственно\(\delta _{\iota - 1} /\delta _\iota \to I\), →μl и →μl 2 приl→∞.
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Sidharth, B.G. On the phase shifts for certain classes of potentials. Nuov Cim A 46, 419–430 (1978). https://doi.org/10.1007/BF02816870
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DOI: https://doi.org/10.1007/BF02816870