Isospectral potentials

Spatially-Dependent-Mass Schrodinger Equations with Morse Oscillator Eigenvalues Isospectral Potentials and Factorization Operators... 

To obtain exactly solvable isospectral potentials on the position-dependent mass problem, let us consider the Morse potential model in the standard Schrodinger-like relationship of Eq.2.7 that means to take as effective potential [9,18]... 

Let us now show the usefulness of the above results by considering a selection of position-dependent mass distributions leading to new isospectral potentials with Morse-type eigenvalues. 

Thus, the exactiy solvable isospectral potential in the corresponding PDMSE... 

This is identical to Eq. (6.45) with A = D and X - 1 = N/2. One also notes that the spectrum of the Poschl-Teller potential in one dimension is identical to that of the Morse potential in one dimension. These two potentials are therefore called isospectral. This identity arises from the fact that, as mentioned in Chapter 3, the two algebras 0(2) and U(l) are isomorphic. The situation is different in three dimensions, where this is no longer the case. 

A standard set of solvable potentials with critical behavior can be found in many text books on quantum mechanics [49,50], like the usual square-well potentials and other piecewise constant potentials. Also there are many potentials that are solvable only at d - 1 or for three-dimensional, v waves like the Hulthen potential, the Eckart potential, and the Posch-Teller potential. These potentials belong to a class of potentials, called shape-invariant potentials, that are exactly solvable using supersymmetric quantum mechanics [51,52], There are also many approaches to make isospectral deformation of these potentials [51,53] therefore it is possible to construct nonsymmetrical potentials with the same critical behavior as that of the original symmetric problem. 

Two points of interest in our algebraic analysis are that (1) this function is symmetrical under 0- -6 (as expected in a bending motion), and (2) it can be shown that for one-dimensional problems, the Poschl-Teller and Morse potentials are isospectral (i.e., they have the same bound-state spectrum) [25]. Figure 4 shows the typical behavior of the Poschl-Teller potential function. 

We now show that the algebraic realization of the one-dimensional Morse potential can be adopted as a starting point for recovering this same problem in a conventional wave-mechanics formulation. This will be useful for several reasons (1) The connection between algebraic and conventional coordinate spaces is a rigorous one, which can be depicted explicitly, however, only in very simple cases, such as in the present one-dimensional situation (2) for traditional spectroscopy it can be useful to know that boson operators have a well-defined differential operator counterpart, which will be appreciated particularly in the study of transition operators and related quantities and (3) the one-dimensional Morse potential is not the unique outcome of the dynamical symmetry based on U(2). As already mentioned, the Poschl-Teller potential, being isospectral with the Morse potential in the bound-state portion of the spectrum, can be also described in an algebraic fashion. This is particularly apparent after a detailed study of the differential version of these two anharmonic potential models. Here we limit ourselves to a brief description. A more complete analysis can be found elsewhere [25]. As a... 

Abstract In this work an algorithm based on the point canonical transformation method to convert any general second order differential equation of Sturm-Liouville type into a Schrodinger-like equation is applied to the position-dependent mass Schrodinger equation (PDMSE). This algorithm is next applied to find potentials isospectral to Morse potential and associated to different position-dependent mass distributions in the PDMSE. Factorization of worked PDMSE are also obtained. 

Fig. 2.1 The principal features of the isospectral exactly solvable potential V(x) given in Eq. 2.25 along with their eigenfunctions and energy spectra, as well as the position-dependent mass distribution. The mass parameter is P = 1/2...

With this definition one has W g x)) = -/3exp(-2/3.x)/2. and the isospectral singular potential of the PDMSE will be... 

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