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 / d x 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 eigenvalua. 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.

Isometric operators are used to provide a unified theory of the three established procedures for generating one-parameter families of isospectral Hamiltonians. All members of the same family of isospectral Hamiltonians are unitarily equivalent, and the unitary transformations between them form a group isomorphic with the additive group of real numbers. The theory is generalized by including the parameter identifying a member of an isospectral family as a new variable. The unitary transformations within a family correspond to translations in the parameter space. The generator of infinitesimal translations represents a conserved quantity in the extended theory. Isometric operators are then applied to the development of models of supersymmetric quantum mechanics. In addition to the standard models based on the Darboux procedure, I show how to construct models based on the Abraham-Moses and Pursey procedures. The formalism shows that the Nieto ambiguity present in all models of supersymmetric quantum mechanics can be interpreted as a renormalization of the ground state of the supersymmetric system. This allows a generalization of supersymmetric quantum mechanics analogous to that developed for systems of isospectral Hamiltonians.

A new procedure is developed for generating families of Hamiltonians which share exactly the same set of eigenvalues. The new method is related to the Marchenko equation in much the same manner as the method of Abraham and Moses [Phys. Rev. A 22, 1333(1980] is related to the Gel'fand-Levitan equation. The two procedures in general yield inequivalent new families of Hamiltonians when used to insert or delete states, but are equivalent (with a proper choice of parameters) when used to renormalize a state. The effect of the new procedure on reflection and transmission amplitudes and on the norming constants for bound states is compared with corresponding results using the Abraham-Moses and Darboux techniques.

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Explicit ~atrix. eleme~~s are foun~ !~~ the ,generators of the ~roup R(5) in an arbitrary irreducible representatIOn usmg the natural basIs m whIch th.e represen.tatIon of R(5) is fully reduced with respect to the.subg~oup R(4) = .SU(~) @ SU(2). ,!,he t~chnIque used IS based on the well-known Racah algebra. The dlmen.slOn formula ~s derIved and the mvarIants are found. A family of identities is established which relates varIOUS polynomIals of degree four in the generators and which holds in any representation of the group.