Polynomial Boundedness Research Articles

Minimal interactions at very high transfers

It is shown that both large-transfer electron-proton scattering and large-angle proton-proton scattering may be consistent with, or close to, the lower limits obtained from dispersion theory. For form factors the fastest allowed decrease is exp[−⇏ |g2|], whereq is the fourmomentum transfer of the photon. For scattering amplitudes one gets exp [−C(θ)⇏s] or exp [−C(θ)√s log s] or exp [−C(θ)√s (log s)2] according to various detailed assumptions, where s is the square of the c.m. energy and θ the c.m. scattering angle. The conditions to impose on the growth of the form factors and the scattering amplitude outside the physical region to obtain these results are considerably weaker than the usual polynomial boundedness condition, and are likely to be satisfied by the actual form factors or scattering amplitudes.

Remarks on the Polynomial Boundedness in the Mandelstam Representation

The Mandelstam representation is a statement about the region of analyticity and asymptotic behavior (polynomial boundedness) of a scattering amplitude. In virtue of the unitarity condition, however, these two are not completely independent. Some physical consequences, e.g., uniqueness, polynomial boundedness of the total cross section, etc., which have been already derived from the Mandelstam representation, are shown to be preserved, even if the polynomial boundedness is replaced by a somewhat weaker assumption. By making use of unitarity, analyticity, and crossing symmetry, the following type of scattering amplitude F = E + M, where E is an entire function in both variables s and t, while M denotes a Mandelstam-type function with finite number of subtraction, is shown to be ruled out. Similarly, F = EM is also ruled out, if one imposes the additional restriction that E should increase less fast than an exponential in one variable while the other is finite.

Connection Between the Asymptotic Behavior and the Sign of the Discontinuity in One-Dimensional Dispersion Relations

Starting from one-dimensional dispersion relations (either fixed-transfer or partial-wave), information on the sign of absorptive part which follows from unitarity, and in some cases analyticity in the Mandelstam ellipse in the $t$ plane and polynomial boundedness, we derive various consequences of physical interest, e.g., a high-energy lower bound on the forward scattering amplitude, the minimum fluctuation of the sign of the discontinuity across the left-hand cut in the partial-wave dispersion relations, etc. In the derivations, the positiveness of the absorptive part plays an essential role by allowing us to construct a Herglotz function which has a well-known asymptotic behavior.

The boundedness of orthonormal polynomials on certain curves of the fourth degree

The boundedness of orthonormal polynomials on certain curves of the fourth degree

The boundedness of orthonormal polynomials on certain curves of the third degree

The boundedness of orthonormal polynomials on certain curves of the third degree

Boundedness of orthonormal polynomials on loci of the second degree

Boundedness of orthonormal polynomials on loci of the second degree