Multi-channel analog of the effective-range expansion

Abstract

Similarly to the standard effective range expansion that is done near the threshold energy, we obtain a generalized power-series expansion of the multi-channel Jostmatrix that can be done near an arbitrary point on the Riemann surface of the energy within the domain of its analyticity. In order to do this, we analytically factorize its momentum dependencies at all the branching points on the Riemann surface. The remaining single-valued matrix functions of the energy are then expanded in the power-series near an arbitrary point in the domain of the complex energy plane where it is analytic. A systematic and accurate procedure has been developed for calculating the expansion coefficients. This means that near an arbitrary point in the domain of physically interesting complex energies it is possible to obtain a semi-analytic expression for the Jost-matrix (and therefore for the Smatrix) and use it, for example, to locate the spectral points (bound and resonant states) as the S-matrix poles.

Combiningab initiocalculations and low-energy effective field theory for halo nuclear systems: The case of7Li+n→8Li+γ

“…The effective-range expansion for the case we studied here, where there are multiple channels with different thresholds, was discussed-albeit in very different formalisms to ours-in Ref. [71,72]; Refs. [73,74] considered effectiverange expansions for multiple open channels with the same threshold.…”

A Method for Extracting the Resonance Parameters From Experimental Cross-Sections

“…The most important limitation of the method described in this paper, is the fact that in its present form the method is only applicable to the systems with short-range interaction forces. A rigorous extension of the method that would include the Coulomb forces, could be done in a way similar to the one described in Ref [16]. This however would require a modified, much more complicated expression for the Jost matrix where all the non-analytic factors (square-root and logarithmic branching points etc.)…”

“…Moreover, in the same Ref. [16] it was established that the matrices A(E) and B(E) are single-valued analytic functions of the energy defined on a single one-layer energy plane. In other words, the matrices A(E) and B(E) are the same for all the sheets of the Riemann surface and all the complications stemming from the branching points are isolated in Eqs.…”

“…In Ref. [16] it was shown that for a given potential the expansion coefficients a n and b n can be obtained as the asymptotic values of the solutions of certain set of differential equations.…”

“…In the present paper, we resolved this difficulty. For obtaining the S-matrix, we use the structure of the Jost matrix derived in our early publications [15,16], where all the factors responsible for the branching are given analytically for an arbitrary number of channels. This enables us to perform the analytic continuation and the search for resonances in a modelindependent way, i.e.…”