Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

1 Answer 1

up vote 1 down vote accepted

The Gelfand-Levitan-Marchenko equation cannot be explicitly solved in general. Only the existence and uniqueness of the solution is proved.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.