Consider a domain $\Omega\_0 \subset \mathbb{R}^n$, and deformations of $\Omega\_0$, called $\Omega\_t$, obtained by a one-to-one mapping $x \mapsto x + t\varphi (x)$, where $\varphi$ is smooth. It is known that the Dirichlet and Neumann eigenvalues of the Laplacian on $\Omega\_t$ vary real analytically with respect to $t$ near $t = 0$. My question is: do the eigenfunctions vary real analytically as well? Any reference would be appreciated. 



Analytic perturbation of eigenfunctions
See this [recent paper](http://www.mat.univie.ac.at/~michor/DC-perturb.pdf) for answers to this and similar questions. You have to translate the parameter dependence of the domain to the parameter dependence of the operator first.


