Stable analytic continuation and unitarity

Abstract

In theoretical physics, in inverting the operator equationAf=h (wheref is «the cause»,h is «the effect» andA is some continuous operator representing a physical law) it is not sufficient forA−1 to exist and to be unique; it is also necessary that it be continuous, in order to preserve the theory from the instabilities that might arise from the never exact knowledge of the inputh. Usually stability is obtained by restricting the set off's in which the solution is sought, to a compact set. The basic idea is then to exploit the fact that different theoretical methods which are logically equivalent (are tautological) and, thus, yield exactly the same result when fed with «absolutely correct» data, could behave in totally different ways when faced with approximate (incomplete and/or error-affected) data. It makes then sense to find among all tautological methods the one which is most insensitive to the imprecision of the knowledge of the input data. In this paper we will first treat the problem of analytic continuation of the scattering amplitude from a limited part of the cuts, first without imposing unitarity and then with imposing it, on the right-hand cut. In Sect.4 we come back to this problem by means of theN/D method, seeking those equations which are less sensitive to the errors of the left-hand data (in this case «correct» data mean that the given left-hand cut limiting values are absolutely consistent with both analyticity and unitarity). Some applications connected with the localization of singularities of the scattering amplitude are then displayed in Sect.5.