The transformations to remove or add bound states for the half-line matrix Schrödinger operator

Abstract: We present the transformations to remove or add bound states or to decrease or increase the multiplicities of any existing bound states for the half-line matrix-valued Schr\"odinger operator with the general selfadjoint boundary condition, without changing the continuous spectrum of the operator. When the matrix-valued potential is selfadjoint, is integrable, and has a finite first moment, the relevant transformations are constructed through the development of the Gel'fand-Levitan method for the corresponding Schr\"odinger operator. In particular, the bound-state normalization matrices are constructed at each bound state for any multiplicity. The transformations are obtained for all relevant quantities, including the matrix potential, the Jost solution, the regular solution, the Jost matrix, the scattering matrix, and the boundary condition. For each bound state, the corresponding dependency matrix is introduced by connecting the normalization matrix used in the Gel'fand-Levitan method and the normalization matrix used in the Marchenko method of inverse scattering. Various estimates are provided to describe the large spacial asymptotics for the change in the potential when the bound states are removed or added or their multiplicities are modified. An explicit example is provided showing that an asymptotic estimate available in the literature in the scalar case for the potential increment is incorrect.