A variational principle for the Gel’fand–Levitan equation and the Korteweg–de Vries equation

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Abstract
A functional is constructed from the spectral density used in the general Gel’fand–Levitan equation and an arbitrary function N of two sets of variables. This functional is shown to be an absolute maximum when N satisfies the Gel’fand–Levitan equation. In the case of the Gel’fand–Levitan equation for the one-dimensional and radial Schrödinger equations and certain generalizations, this result can be translated into a theorem about the area under a curve to a given point (x or r), considered as a functional of N. This curve is given by the scattering potential to the given point when the functional takes on its maximum value. The functional may thus be considered a method of obtaining the scattering potential from the spectral data through a variational technique. In the case that the Gel’fand–Levitan equation is that for the one-dimensional Schrödinger equation the results can be interpreted as a theorem about the area to a given point x under the curve given by the solution of the Korteweg–de Vries equation. That is, at a given time t the area under a curve to a given point x, considered as a known functional of N, takes on its maximum value for all x and t when the curve represents the solution of the Korteweg–de Vries equation with appropriate initial conditions.