New Schrödinger equations for old: Inequivalence of the Darboux and Abraham-Moses constructions

There exist two methods for generating families of isospectral Hamiltonians: one based on a theorem due to Darboux and the second due to Abraham and Moses based on the Gel’fand-Levitan equation. Both methods start with a general Hamiltonian operator H=-$d^2$/${\mathrm{dx}}^2$+V(x), and generate infinite families of new Hamiltonians all with the same eigenvalue spectrum. The new spectrum corresponds either to the addition of new bound states with specified energy eigenvalues or to the deletion of bound-state eigenvalues. Neither process (addition or deletion) alters the reflection or transmission probabilities, although the amplitudes experience a phase change consistent with Levinson’s theorem and the change in the number of bound states. In this paper we show that these two methods of generating families of isospectral Hamiltonians are, in general, inequivalent.