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Extra info for Advances in Cryptology: Proceedings of EUROCRYPT 84 A Workshop on the Theory and Application of Cryptographic Techniques Paris, France, April 9– 11, 1984

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Thus we have a situation similar to that in Fig. 3, except that all the 77^ for 1 will be extremely small over a large energy range, in which may drop through at least several of its VQ multiples of π. This yields a series of minima and maxima in the total elastic cross section corresponding to taking integral and half-integral multiples of ττ, respectively. 26 4. 2) resonance state, fi{±ß + iy) = 0, with β,γ > 0. 3) It should be emphasized that whether or where such zeros occur on the complex k plane will depend on the solutions to the radial wave equation for the particular potential field of interest.

25) in the Hmit of ák-^O, With no potential present dηι|dk = 0, and ^max — -for t ^ 0 describes the position of maximum amplitude in the outgoing wave packet. 26) is the time delay experienced by the wave packet in scattering, as com­ pared with the case of no potential field, when {At)i = 0. 2), the time delay may be expressed as {At)i = -ifi (dSildE) s r . 27) This latter definition is the general time-delay matrix for the manychannel case when the appropriate many-channel S matrix is used. 34 5a.

Thus Si^k) = 0 when I + I — ioc = —ñ with ñ = 0, 1, 2,... Using the definition of a, these occur at . = . - < ^ , which, for an attractive Coulomb field (ZZ' < 0), lie on the negative imaginary axis of the k plane. These zeros of Si^ correspond to the infinite set of bound energy levels in an attractive Coulomb field. 44) where η = / + 1 + ñ is the principal quantum number. Thus the zeros of Si^ correspond to bound states in the same way as for the usual S matrix when dr r \ U \ is finite. However, Levinson's theorem is not apphcable to the Coulomb case since rfr r | U^l = oo, and should not be regarded as a particular form of the non-Coulomb ηι.

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