Let $f$ be a positive real-analytic function on the closed unit disk. Consider the eigenvalue problem $\Delta \phi = \lambda f \phi$, with $\phi = 0$ on the boundary. There exists a sequence of eigenvalues $\lambda_n$. Now suppose $f$ depends real-analytically on a parameter $t$ for $t$ in some interval containing $0$. Let $n$ be given and let $k$ be the dimension of the eigenspace of $\lambda_n$. Do there exist $k$ functions of $t$, defined on some interval about $t=0$, that are at least $C^1$ in $t$ and give $k$ eigenvalues of $t$, all of which equal $\lambda_n$ when $t=0$?
Remarks: (1) Courant-Hilbert, Methods of Mathematical Physics, vol. 1, page 419 proves continuity (i.e. $C^0$ dependence). You would think if $C^1$ dependence were known (in 1953) they would mention it.
(2) We do not require that the functions preserve the order of the eigenvalues. For example it might happen that $\lambda_2 < \lambda_3$ for $t < 0$ and $\lambda_2 > \lambda_3$ for $t > 0$, and maybe they cross transversally, though I do not know if this can really happen.
(3) Perhaps one can even get real-analytic dependence on $t$.
(4) There is a thick book by Kato on perturbation theory with a lot of theorems in it, one or more of which possibly contains the answer.