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We develop an extension of a class of degenerate Gel'fand-Levitan and Marchenko equations previously used in the theory of isospectral Hamiltonians and in the theory of continuum bound states. The formalism allows the generation of new exactly solvable Schrodinger equations, whose potentials extend the class of Bargmann potentials. The scattering theory is developed using the Jost solutions of the new Schrodinger equations, leading to exact expressions for the Jost function and the scattering phase shifts. The theory is illustrated by a simple example.

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No abstract