Eigenvalues and eigenfunctions for the Fredholm integral operator $K(g) = \int_0^1 e^{x t} g(t) dt$.

I would like to compute the eigenvalues and eigenfunctions for the Fredholm integral operator

$$K(g) = \int_0^1 e^{xt} g(t) dt.$$

The sources I've checked* seem to say that the process is fairly involved. Has anything been published on this kernel? Or, if not, am I correct that it's going to be a hard thing to do?

* See, e.g., equations (12) and on here: https://www.encyclopediaofmath.org/index.php/Fredholm_equation

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$k$ is operating on which space? – Martin Argerami Feb 13 '12 at 5:16
@MartinArgerami: $L^2(\mathbb{R})$. – Antonio Vargas Feb 13 '12 at 5:33
In that case I would expect your case to be harder than those in the link you provided: because they require the kernel to be in $L^2$ of the product space, while yours isn't. – Martin Argerami Feb 13 '12 at 15:54
@MartinArgerami: Sorry, I should have been more clear, I meant $L^2([0,1])$. The kernel of the operator in question, $e^{xt}$, is certainly in $L^2([0,1]^2)$, and the operator acts on $L^2([0,1])$. The link definitely applies here. – Antonio Vargas Feb 13 '12 at 17:02