New families of isospectral Hamiltonians

Abstract

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.

Electrons above a helium surface and the one-dimensional Rydberg atom

“…A first examination of the above inverse methods [27]- [30] found potentials with the last two properties, but not the first. These potentials go to zero at the origin.…”

Extended Gel’fand-Levitan method leading to exactly solvable Schrödinger equations with generalized Bargmann potentials

“…[9]. [7,8] and to the theory of isospectral Hamiltonians [3,4]. The development presented here provides yet a further extension of the class of exactly solvable Schrodinger equations.…”

“…However, the extended formalism which we present here far exceeds this goal. Indeed, the formalism permits the creation of a class of exactly solvable Schrodinger equations similar to but much more extensive than those generated by the theory of isospectral Hamiltonians [3,4].…”

“…Moses and Tuan [10] used the Gel' fand-Levitan equation to create potentials which produced no scattering phase shift, and incidentally discovered a procedure for generating bound states with en-ergies embedded in the continuum. The Gel'fand-Levitan equation has also been used by Meyer-Vernet [11]to study continuum bound states, while we [12] [13].For the same system, Luban and Pursey [3] compared the Gel'fand-Levitan-Abraham-Moses system of isospectral Hamiltonians with systems developed using the Darboux trick [14] (best known today as the basis of supersymmetric quantum mechanics [15]), while Pursey [4] [1], and readers are referred to that paper for details.…”

Information Entropy of Nanostructure systems

“…Using these concepts, the isospectral Hamiltonian method has been utilized in the generalization of the solutions for different systems [13][14][15][16][17]. The energy spectrum of potential is derived upon utilizing the relationship in eigenfunctions and the potential.…”