Finite-rank potential that reproduces the Padé approximant
Smio Tani · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · 1979
If a scattering potential is of a finite rank, say $N$, the exact solution of the problem can be obtained from the Born series, if the potential strength is within the radius of convergence; the exact solution can be obtained from the analytical continuation of the formal Born series outside the radius of convergence. Beyond the first $2N$ terms of the Born series, an individual term of the Born series depends on the first $2N$ terms, and the [$\frac{N}{N}$] Pad\'e approximant and the exact solution agree with each other. The above-mentioned features of a finite-rank problem are relevant to scattering theory in general, because most scattering problems may be handled as an extension of the rank-$N$ problem, in which the rank $N$ tends to infinity. The foregoing aspects of scattering theory will be studied in depth in the present paper, and in so doing we proceed in the opposite direction. Namely, given a potential, we calculate the first $2N$ terms of the Born series for the $K$ matrix and the first $N$ terms of the Born series for the wave function. Using these data, a special rank-$N$ potential is constructed in such a way that it reproduces the [$\frac{N}{N}$] Pad\'e approximant of the $K$ matrix of the original scattering problem. One great advantage of obtaining such a rank-$N$ potential is that the wave function of the system may be approximated in the same spirit as done for the $K$ matrix; hence, we can introduce a new approximation method for dealing with an off-shell $T$ matrix. A part of the mathematical work is incomplete, but the physical aspects are thoroughly discussed.