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Given the Sturm-Liouville operator

$$ - \frac{d^{2}}{dx^{2}}y(x)+y(x)q(x)=zy(x),$$

my question is how to use spectral data to obtain $ q(x) $ inside the last equation by the Gelfand-Levitan-Marchenkio method

$$ q(x)= 2 \frac{d}{dx}K(x,x)$$

for the case of this problem with even potential $ q(x)= q(-x) $ or on the half line $ [0. \infty) $. I know how to get $ q^{-1}(x) $ but i would be more interested in getting $ q(x) $ instead.

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Reconstruction of the potential of the Sturm-Liouville problem by spectral data is too long to describe here. You can study it from the following books:

  • Levitan B.M. Inverse Sturm-Liouville problems. VSP, Zeist, 1987. x+240 pp.
  • Marchenko V.A. Sturm-Liouville operators and applications. Birkhäuser Verlag, Basel, 1986. xii+367 pp.
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