Search for physical structures on the boundary by optimal analytic continuation from a finite set of interior data points

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Abstract
The object of this paper is to develop a method for obtaining information about the discontinuity function along the cuts (which is related to the positions and widths of resonances), from data—either experimental or theoretical—given at some points inside the holomorphy domain. This will be achieved by means of an analytic continuation which is required to be optimal under some specific boundary conditions. Once errors are present or the number of data points is finite, analytic continuation is no longer unique but highly unstable. To give a well defined continuation prescription, a stabilizing condition is essential, and the latter has to be chosen to suit the physical problem under consideration. It is shown how such a continuation procedure may be used (a) to ascertain whether the data can be said to require a particular type of structure on the boundary such as that which would arise from a nearby pole on the second Riemann sheet, as would be associated with a resonance, and (b) if so, to determine the parameters of such a resonance. Among the many applications of the method derived in this paper, one which is of some topical interest, is to use as input the result of perturbative calculations in some region of the complex plane where such expansions may be meaningful (e.g., asymptotic or negative energy in QCD) and to attempt to compute quantities of physical interest in a region where direct perturbative calculations are not valid.

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