Approximate Solution to the MultichannelND−1Equations

By making a pole approximation to the spectral integral over the kinematical factor ${\rho }_l\mathrm{}\left(z\right)\mathrm{}$ it is shown that the partial-wave matrix $N{D}^{-1\mathrm{}}$ integral equations are reduced to algebra. The approximation depends only on the particular partial wave and not the dynamics of the reaction, and it admits of systematic improvement. The resulting scattering amplitude $T_l\mathrm{}\left(z\right)\mathrm{}$ is symmetric, is independent of the subtraction point for the $D$ function, has the correct discontinuities on the right- and left-hand cuts, and can moreover be explicitly expressed as an algebraic function of the driving term $B_l\mathrm{}\left(z\right)\mathrm{}$. This last feature enables us to directly inspect the relation between the driving force and the scattering amplitude, and establishes the general usefulness of the method. We find, for example, that the solution imposes general conditions on $B_l\mathrm{}\left(z\right)\mathrm{}$ for the existence of bound states, resonances, or possible ghosts. The self-consistency property of bootstrap calculations imposes additional explicit restrictions on acceptable $B_l\mathrm{}\left(z\right)\mathrm{}$ for the existence of the bootstrap.