Jost function for the description of resonance in finite quantum multichannel systems

The Jost function is defined as the coefficient function connecting the regular and irregular solutions of the fundamental differential equations. It is known that the zeros of the Jost function on the complex energy plane correspond to the poles of the S-matrix, which represent the complex eigenvalues of bound and resonant states. This paper reviews our recent extensions of the Jost function method to nuclear multichannel systems, specifically within the frameworks of the Hartree-Fock-Bogoliubov (HFB) theory and the Random Phase Approximation (RPA) theory (Jost-RPA method). A unitary S-matrix is derived using these extended Jost functions. By focusing on the poles of the S-matrix, we attempt to analyze and classify the resonances. We discuss three key applications: (1) the extraction of Fano parameters to analyze asymmetric line shapes in neutron scattering within the HFB framework, (2) the decomposition of the RPA strength function using eigenphase shifts, and (3) the application of the Mittag-Leffler theorem to decompose the RPA response into contributions from individual resonance poles.