I need to determine the eigenfunctions of the following kernel:
$$k\left(x',x\right)=\frac{1}{\sqrt{2\pi ax}}\exp\left(-\frac{\left(x'-x\right)^{2}}{2ax}\right)$$
where $a>0$ is a positive real parameter and $x$ goes from $0$ to $1$. Specifically, I need to determine the eigenfunctions with eigenvalue 1. That is, I need to find the functions $f(x)$ that are solutions of the following equation:
$$f(x') = \int_0^1 k(x',x)f(x)\mathrm d x$$
Note that the kernel is not symmetric, $k(x,x') \ne k(x',x)$. Also, we need to impose a regularity condition at $x\rightarrow 0$. We require that $|k(x',x)f(x)|^2 \le A$, where $A$ is a positive constant.
The "kernel" you wrote is not integrable even for $L^2$ functions on the positive real line. To see this let $x'=0$ and $f\sim |x|^{-1/4-\epsilon}$ near zero, then you need to integrate $$ \sim \int \frac{1}{x^{1+\epsilon}}e^{-x} $$ and near $x=0$ we have a blow-up. So the formula makes little sense unless you impose certain decay conditions. I suppose the formula comes from certain exponential scale-location family.