Analytic solutions, Darboux transformation operators and supersymmetry for a generalized one-dimensional time-dependent Schrödinger equation

Abstract In this paper, analytically investigated is a generalized one-dimensional time-dependent Schrodinger equation. Using Darboux transformation operator technique, we construct the first-order Darboux transformation and the real-valued condition of transformed potential for the generalized Schrodinger equation. To prove the equivalence of the supersymmetry formalism and the Darboux transformation, we investigate the relationship among first-order Darboux transformation, supersymmetry and factorization of the corresponding effective mass Hamiltonian. Furthermore, the nth-order Darboux transformations are constructed by means of different method. Finally, by using Darboux transformation, some analytical solutions are generated in a recursive manner for some examples of the Schrodinger equation.

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