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. 



dependence of eigenvalues on parameters
In your case I think you can apply Rellich's theorem, that is Theorem VII.3.9 in Kato's book (p.392 in my edition). The result states that, whenever you have a family of selfadjoint operators with compact resolvent, depending analytically on a real parameter on some open interval of the reals, with a common domain independent of the parameter (this is what Kato calls a family of type (A)), then you can enumerate both eigenvalues and eigenfunctions in such a way that they are analytic functions of the parameter in the same interval.


