Mixed procedures for generating families of isospectral Hamiltonians

Three standard procedures for generating families of isospectral Hamiltonians, either by introducing a new ground state or by deleting the original ground state, are used in combination. I investigate the unitary transformations resulting either from using one process to insert a new state and a different process to remove it again (denoted symbolically by $Z^{\mathrm{°}}$Y), or from using one process to remove the original ground state and a different process to reintroduce a state with the original ground-state energy (denoted symbolically by ${\mathrm{ZY}}^{\mathrm{°}}$). Many connections are found between the resulting procedures, all of which are related to just two one-parameter families of unitary transformations, $U_{\epsilon }$ associated with the $Z^{\mathrm{°}}$Y processes and $T_{\xi }$ associated with the ${\mathrm{ZY}}^{\mathrm{°}}$ processes. The unitary transformations $Z^{\mathrm{°}}$Y are found to be limiting cases of the nonunitary isometric operators associated with procedures for inserting a new ground state.