Applications of the Gel’fand-Levitan Equation
The Gel’fand-Levitan equation being a Fredholm equation, it is obvious that it cannot be solved in general in closed form. Except in a very few exceptional cases, the most general case solvable in closed form is when the kernel is degenerate. In this case, it is known that one can solve the equation algebraically. The most interesting instances of this situation are the following : In the first one, we start from a potential V1 which has no bound states, and for which everything (wave function φ1, spectral density dρ1(E)) is known, and we would like to find all the potentials V which have the same spectral density for positive energies (the same continuum) but have, in addition, some bound states. The second case is when we start from a given Jost function, assuming again that all quantities related to it, including the potential, are known, and would like to find all the potentials whose Jost functions differ from the first one by a rational factor of k. We leave this problem for the third section, and now study the first problem.
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