## Homogeneous Fredholm equation with symmetric Kernel

Hello, I'm trying to calculate the Schmidt decomposition of an entangled quantum state, and I stumbled across an homogeneous Fredholm equation of the second kind: $$u_k(x)-\lambda_k\int_1^\infty K(x,y)u_k(y)dy=0$$ The kernel contains a rational function, a Gaussian function and a Bessel function in this fashion: $$K(x,y)=\sqrt{xy}\exp[-c(x^2+y^2)]I_m(xy)$$ My question is: is there a general method to find the eigensystem for a symmetric nonseparable kernel? Could anyone point me to some references? I found a lot of material on properties of symmetric kernels and on their eigensystems, but no methods to actually find them. (I would love to turn it into a differential equation, for instance.)

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 Update: I found a solution from an old article by Hardy and Hille cited in Watson's J. London Math. Soc. 1933 189-192! :D – Filippo Jul 27 2011 at 10:24