Resonances in three-body scattering theory (Q1822023)

In this paper resonances in multichannel scattering theory associated with the three body problem for negative energies, assuming exponential decay of the two-body interactions, is discussed. These resonances are characterized as (1) poles of an analytically continued resolvent acting between certain exponentially weighted spaces, (2) eigenvalues of a Schrödinger operator acting in a suitable space, (3) singular points of the Faddeev matrix, (4) singular points of the Lippman-Schwinger operator, (5) poles of the S-matrix, (6) poles of the analytic families of the exponentially growing eigenfunctions. The analysis is restricted to the analytic continuation of the resolvent, the S-matrix etc., across the spectrum to the second sheet in the lower and upper half-planes and the resonances in this region. Further how the various operator- and vector-valued functions can be analytically continued on a Riemann surface determined by two sheeted parabolic regions focused at thresholds as \(\sqrt{z}\)-type branch points is indicated.