Abstract. Let q(r) , r = |x| , x ∈ R 3 , be a real-valued square-integrable compactly supported function, and [0, a] be the smallest interval containing the support of q(r) . Let A(α , α) = A(α · α) be the corresponding scattering amplitude at a fixed positive energy, k 2 = 1 . Let δ be the phase shifts at k = 1 . It is proved that lim = a , provided that q(r) does not change sign in some, arbitrary small, neighborhood of a .

In this work we study the Cosine Transform operator and the Sine Transform operator in the setting of Henstock-Kurzweil integration theory. We show that these related transformation operators have a very different behavior in the context of Henstock-Kurzweil functions. In fact, while one of them is a bounded operator, the other one is not. This is a generalization of a result of E. Liflyand in the setting of Lebesgue integration.

New sufficient conditions for the existence of generalized Hopf bifurcations are given in the context of asymptotically compact dynamical or semidynamical systems on a metric space. These conditions weaken the hypotheses of previous contributions to the subject.

Abstract. In this paper we study the impact-parameter model for the scattering of a light particle by two heavy ones in the case when the coupling constants of the potentials acting on the light particle due to the presence of the two heavy ones are the same. We study the asymptotic behavior of the transition probability when the relative velocity of the heavy particles goes to zero. We show that the probability of a transition can be arbitrarily close to the one of no transition. Section 1In this paper we consider the impact-parameter model for the scattering of a light particle by two heavy ones. We continue the study of the asymptotic behavior of transition probabilities in this model. In a previous paper [1] we showed that the transition probability at low velocities from an initial bound state tends to zero for unequal coupling constants. In the case of equal coupling constants, there exists the possibility of eigenvalues arbitrarily close to each other, each of them equal probable. Therefore the probability of a transition should be almost equal to the probability of no transition. We give in this paper a rigorous justification of this statement.Let L2(R m) denote the Hilbert space of Lebesgue-measurable square-integrable complex-valued functions, defined on R m. For t e R 1, let Hv(t) be the following self-adjoint operator in L 2 (R") with domain H 2 (R"), the Sobolev space of order two with derivatives in distribution sense. 2 is real, 2 > 0. Let (., .) denote the scalar product in L2(R ") antilinear in the factor on the left. For g ~ L2(R m) we define the rank-one operator Vp,

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