Assume we have a complete orthogonal system on a domain $D$, given by the eigenfunctions of the Laplacian on $D$. For example, the set $\{e^{int}\}$ on $[-\pi, \pi]$, or the spherical harmonics on the unit ball. Now consider a domain $D'$, which is "close" to D in some sense (the boundary of $D$ is close to the boundary of $D'$ in some suitable norm).

Are the eigenfunctions of the Laplacian on $D$ close, in some sense, to the eigenfunctions of the Laplacian on $D'$? Does knowing the basis of $D$ help approximate the basis of $D'$ ? Any known results along these or similar lines appreciated.