Lower Bounds on the Inverse Reactance Matrix

The spectrum of the closed-channel Hamiltonian $\mathcal{H}$ in the multichannel scattering problem is bounded from below and by construction is discrete in the energy region below the threshold for new channels. This property has been applied previously to derive a minimum principle which provides an upper bound on the inverse reactance matrix. The discreteness of the spectrum of $\mathcal{H}$ in the region below the threshold for new channels is further exploited here to derive lower bounds. Forms linear as well as quadratic in $\mathcal{H}$ are obtained and their applicability discussed. Extensive use of the Unsöld approximation and the closure properties of the states generated by the various operators is necessary in order to reduce the inequalities to manageable forms. An iterative method to treat the energy shift operator is discussed, and the convergence of the iteration series and its connection to the subtraction terms in the minimum principle are discussed in detail.