While it might sound that this question is all too familiar, I've been struggling finding references about this. Suppose that I have a square integrable kernel $K(x,y)$ such that $K$ is symmetric, i.e. $K(x,y) = K(y,x)$. Assume that $x,y$ are in a bounded domain. I am interested in the eigenvalue problem $\int K(x,y) \phi_k(y) \,dy = \lambda_k \phi_k(x)$ where $\phi_k$ are the eigenfunctions while $\lambda_k$ are the eigenvalues.
More specifically, I am looking for bounds on the eigenvalues and eigenfunctions should there be a perturbation of $K(x,y)$, i.e. assume that $\tilde{K}(x,y) = K(x,y) + e(x,y)$ where $e(x,y)$ is nicely behaved (i.e. regularity conditions) and that $e(x,y)$ is also symmetric. As $K$ is perturbed, it is expected that $\lambda_k$ and $\phi_k$ will also be perturbed and I am trying to find bounds on the discrepancy between $\tilde{\lambda_k}$ and $\lambda_k$ as well as $\tilde{\phi_k}$ and $\phi_k$ where $\tilde{\lambda_k}$ and $\tilde{\phi_k}$ are the eigenvalues/functions of $\tilde{K}.$
My reference starting point is this article on wikipedia on eigenvalue perturbation for matrices, i.e. the finite dimensional case. (link here). While it seems to be that the technique presented there can be adapted in a straightforward manner in the infinite dimensional case (i.e. for kernels), I am bothered by one line in the derivation which I think is a bit hand-wavy, the part which says: "Removing the higher-order terms". Is there a better way to go about it?
Are there any other references that deal with this for the kernel case that are citable? I've been looking for books and references but seem to be typing the wrong key words.
Thanks for the suggestions!