The foundation of the inverse spectral problem were laid down some 50 years
ago by Gelfand and Levitan for the Sturm-Liouville operator -y'' +qy and by
M. G. Krein for the string defined by the differential expression
dy'/dM(x). By using Stieltjes continued fractions and the moments problem,
in some cases MG Krein could recover the mass of the string M, explicitly.
On the other hand, the Gelfand-Levitan theory, utilizing very simple tools
such as Fredholm integral equations, gave a simple algorithm leading to the
recovery of the potential q. The main drawback was a restricted class of
spectra in comparison with the string. Due to the importance of the
Schr¨odinger equation numerous numerical methods based on the
Gelfand-Levitan theory and integral equations have been tried out. In
general, one has to solve complicated nonlinear systems by iterative or
Newton methods.
Assuming that all we want is a polynomial approximation of the potential q,
can one develop a new and direct method that is simple enough to be
implemented on a small machine? We show that the simplicity of the power
series expansion allows us to achieve this goal at a very low cost. Recall
that in order to recover q we must either be given a spectral function or
two spectra. For the sake of simplicity we use the latter as it coincides
with the regular case. The issue related to the need for two spectra is made
apparent in the solution of the inverse problem. It is our hope that working
with ideas from function theory will help bring about efficient
computational methods to solve the inverse spectral problem.
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