Mixed procedures for generating families of isospectral Hamiltonians.

Abstract

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 ${Z}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$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 ${\mathrm{ZY}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$). Many connections are found between the resulting procedures, all of which are related to just two one-parameter families of unitary transformations, ${U}_{\ensuremath{\epsilon}}$ associated with the ${Z}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$Y processes and ${T}_{\ensuremath{\xi}}$ associated with the ${\mathrm{ZY}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ processes. The unitary transformations ${Z}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$Y are found to be limiting cases of the nonunitary isometric operators associated with procedures for inserting a new ground state.Several limiting cases of the new unitary transformations yield nonunitary transformations associated with deletion of the ground state. For a scattering system, I present the effects of the unitary transformations ${U}_{\ensuremath{\epsilon}}$ and ${T}_{\ensuremath{\xi}}$ on the transmission and reflection coefficients and on the norming constants of the bound states. The unitary operators ${U}_{\ensuremath{\epsilon}}$, ${T}_{\ensuremath{\xi}}$, and their inverses are shown to intertwine in a natural manner with each other and with the unitary transformations ${W}_{\ensuremath{\xi}}$ defined in an earlier paper [D. L. Pursey, Phys. Rev. D 33, 2267 (1986)]. In particular, I show that ${T}_{\ensuremath{\xi}}$=${W}_{\ensuremath{\xi}}$${T}_{0}$, and that iteration of T\ensuremath{\equiv}${T}_{0}$ generates a group of transformations isomorphic with the additive group of integers. This group is a symmetry group of a generalized formalism of the theory. This extended formalism is also invariant under a group isomorphic with the group of rotations in a plane. When either the group generated by T or the associated two-dimensional rotation group is combined with the one-parameter Lie group of transformations associated with the operators ${W}_{\ensuremath{\xi}}$, the intertwining relations show that the resulting group is a direct product. I conclude with a brief survey of remaining problems and of possible further generalizations of the theory.