Inverse spectral problem: How to recover the function $ q(x) $?

Asked Dec 01 '12 at 11:59

Active Jan 21 '17 at 11:18

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The forward problem is a second order Sturm-Liouville operator

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

with the boundary conditions $ y(0)=0=y(\infty) $.

If I know the spectral measure function $ \sigma (x) =\sum_{\lambda_{n} \le x}1 $, then can I reconstruct the inverse of the potential $ q^{-1}(x) $?

My question is, how do I use the Gelfand-Levitan-Marchenko theory to reconstruct the potential $ q(x) $?

edited Jan 21 '17 at 11:18

Tommi

asked Dec 01 '12 at 11:59

Jose Garcia

If the potential does not have any bound states, it will have the same spectral measure as the zero potential, isn't it? – Appliqué Jun 19 '19 at 12:34

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