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Nonrelativistic S‐Matrix Poles for Complex Angular Momenta
1.T. Regge, Nuovo Cimento 14, 951 (1959).
1.Added in proof. In view of some confusion in the references to the historical origin of the replacement of this series by a contour integral it may be well to straighten out the record. The method is based on those introduced in the context of bending of electromagnetic waves by a sphere by H. Poincaré [Rendiconti Circolo Mat. Palermo 29, 169 (1910)]
1.and J. W. Nicholson [Phil. Mag. 19, 516 (1910);
1.J. W. Nicholson, 20, 157 (1910); , Philos. Mag.
1.J. W. Nicholson, Messenger Math. 37, 84 (1907)].
1.However, the exact form of the general method as it is used at present first appeared in the work of G. N. Watson [Proc. Roy. Soc. (London) 95, 83 (1918)].
1.A. Sommerfeld [Partial Differential Equations in Physics (Academic Press Inc., New York, 1949), pp. 282 ff.] deserves credit for resurrecting it. I am indebted to Dr. E. Guth for bringing the Poincaré reference to my attention.
2.T. Regge, Nuovo Cimento 18, 947 (1960).
3.A. Bottino, A. M. Longoni, and T. Regge, Nuovo Cimento 23, 954 (1962).
4.G. F. Chew, S. C. Frautschi, and S. Mandelstam, Phys. Rev. 126, 1202 (1962).
5.The width of the resonance thus depends not only on the nearness of the pole to the real axis but also on its energy dependence. Equation (1.8) differs from (1.6) of reference 4 which failed to take the energy dependence of Im α into account. It should be remembered in addition, as in all resonance theories, that to speak of a “resonance” is observationally meaningful only if it is sharp relative to other energy dependencies in the amplitude. Otherwise the resonance denominator need not lead to a maximal partial cross section. Furthermore, the energy shift must also be small in order for the result to be meaningful. If is not small then no resonance occurs near and near the neglected terms in the expansion of become important.
6.The following is entirely analogous to what is done for integral values of l and closely follows the procedure of reference 7.
7.R. G. Newton, J. Math. Phys. 1, 319 (1960).
8.From a semi‐classical point of view the inequality (3.3) is well known. Since the classical deflection function Θ is connected with the WKB phase shift by (3.3) merely states the fact that (“Orbiting” implies a large negative Θ.) I am indebted to Dr. Joseph W. Weinberg for calling the connection with the classical result to my attention.
9.N. Levinson, Kgl. Danske Videnskab. Selskab, Mat.‐Fys. Medd. 25, No. 9 (1949).
10.In other words before
11.It should be noticed that the foregoing result refers directly to observable resonances in the sense that and not merely to S‐matrix poles, which may or may not entail observable resonances.
12.Since the right‐hand side is positive, this proves the physically obvious fact that a zero on the real positive axis always moves to the right as the energy increases. The expectation value of depends, of course, both on k and on λ, and (4.3) must not be mistaken for a constant. This equation was first obtained by Regge. However, in contrast to his use of (5.1) in reference 2, it should be remembered that (4.3) cannot be used when since then φ is not normalizable.
13.The result will hold under much more general conditions, but for a potential that vanishes asymptotically as a power of r, it breaks down if at is too large.
14.These restrictions do not hold for Re though. Added in proof: Numerical work was meanwhile shown that for a Yukawa potential, zeros occur both in the third and in the fourth quadrant; see A. Ahmadzadeh, P. G. Burke, and C. Tat (to be published).
15.Again the result holds under much more general conditions.
16.After this work was finished, the author learned that Barut and Zwanziger have come to similar conclusions; A. O. Barut and D. E. Zwanziger (to be published).
17.The present derivation of this result implies that the l value in it is not that of the resonance but that at which the trajectory meets the real axis at If that happens at then but it is then quite unlikely that the trajectory causes a p‐wave resonance at sufficiently low energy for the formula to be applicable. In general it is not likely to be applicable unless the l values of the resonance and of the point are nearly equal.
18.This suggestion appears to be contradicted by the fact that as Re The resolution of this quandary is presumably that the turning point of a trajectory cannot have a small imaginary part. If it does not, then the shift term is not small and the resonance formula has no significance. See footnote 5.
18.Added in proof. In the square well case it is now known not to do so; see A. D. Barut and F. Colagero (to be published).
19.R. Blankenbecler and M. L. Goldberger (to be published).
20.M. Froissart, private communication from M. L. Goldberger.
21.A. Ahmadzadeh, P. G. Burke, and C. Tate (to be published).
22.V. Singh (preprint).
23.For the purpose of this argument, it is sufficient that the potential be negative somewhere. We may then multiply the attractive piece by a positive parameter and increase it.
24.V. Bargmann, Proc. Natl. Acad. Sci. U.S. 38, 961 (1952).
25.Bargmann has shown that for fixed l, (5.1) cannot be improved without special restrictions on the potential. For given l there always exists a potential that causes (5.1) to be as near to equality as one pleases. Here, however, we are interested in improving (5.1) if we fix the potential and let l approach There is no contradiction.
26.J. Schwinger, Proc. Natl. Acad. Sci. U.S. 47, 122 (1961).
27.It must be remembered that for and small λ the eigenvalue criterion is quite weak. The demand is merely that the solution which is dominant (“irregular”) at also be dominant at That implies that the solution which is more “regular” at the origin is also more “regular” at infinity. “More regular” here simply means “asymptotically smaller.”
27.Added in proof. S. C. Frautschi, M. Gell‐Mann, and F. Zachariasen, Phys. Rev. 126, 2204 (1962) have come to the same conclusion.
28.It can be shown directly for an attractive square well potential that, as the depth tends to naught, the angular momentum of the eigenvalue tends to
29.In addition, of course, to the general assumptions that are necessary for the considerations of Re
30.In the following N always stands for a positive integer, unless otherwise indicated.
31.Strictly speaking, contains a logarithmic term so that appears to be of order The ln term generally cancels, though. Whether it is there or not is of no consequence in the following.
32.Some or all of these poles may be absent in special cases. For example, if V remains bounded as then there is no pole at For a square well potential of radius R, φ has no pole for it is essentially a Bessel function.
32.Added in proof. Meanwhile, numerical work by A. O. Barut and F. Calogero (to be published) indicates that for repulsive square‐well potentials, complex negative energy zeros do occur. This will be discussed in more detail in a forthcoming publication with B. R. Desai.
33.It would be a mistake to believe that such a coincidence of zeros and poles at may be a result of the anomalous tail of the Coulomb field. It must happen, for example, for the Yukawa potential too. Otherwise there would be a prohibition against a pole moving through for large enough N.
34.This is Eq. (4.4) of reference 7.
35.The integral in (8.4) does not diverge as because we have seen that there the first terms in φ(or ζ) have simple zeros and ζ goes as
36.Nothing is known that would prohibit one end of a trajectory from being at and the other at or at
37.It was noted at the end of Sec. 6 that the proof of the meromorphic character of S can be extended to cases in which rV goes as say, near the origin. It is clear in view of the foregoing considerations that in such a case the ends of trajectories may lie at nonintegral values of The possible end points of trajectories are the poles of φ, i.e., of F. These are easily obtainable from α.
38.This was also shown directly by B. R. Desai (private communication). It is clear from our general reason that this is not due to the sharp cutoff at R. For any potential that is constant over a finite region starting at the trajectories must end at infinity in the λ plane. In that case φ has no poles.
39.This was shown also in reference 3.
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