# Gelfand-Levitan-Marchenko equation

how can one solve the integral

$$f(x,y)+K(x,y)+\int_{0}^{x}K(x,t)f(t,y)dt =0$$ (1)

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

$$-y''(x)+q(x)y(x)=0$$ (3)

$$y(0)=0=y(\infty)$$

$q(x)$ here is the pontential of a Sturm Liouville operator (3)

$$\phi (x) = \int_{-\infty}^{\infty}\frac{d\lambda}{\lambda}(1-cos(\sqrt{x}t)\rho (\lambda)$$ (4)

$$f(x,y)= \frac{ \partial _{x}^{2}\phi (x+y)+\partial _{x}^{2}\phi (x-y)}{2}$$

here i have a doubt, inside the Gelfand-Levitan equation what is $\rho (x)$

also is there an asymptotic or analytic solution to this equation ?? thanks.

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