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3 added "sp.spectral-theory" tag
2 edited for clarity

Assume we have a complete orthogonal system (with respect to the L2 norm) on a domain $D$ (for e.g.D$, given by the eigenfunctions of the Laplacian on$D$. For example, the set${e^{int}}$on$[-\pi, \pi]$, or the spherical harmonics on the unit ball)ball. Now consider a domain$D'$, which is "close" to D in some sense (say, the L2 distance between the boundary of$D$and is close to the boundary of$D'$is upper bounded)in some suitable norm). What can be said about Are the basis eigenfunctions of the Laplacian on$D'$? Is there any sense D$ close, in which some sense, to the functions in eigenfunctions of the Laplacian on $D'$? Does knowing the basis of $D$ are "close" to the functions in help approximate the basis of $D'$ ? Any known results along these or similar lines appreciated.

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# how does the basis of an inner product space change when the domain is deformed

Assume we have a complete orthogonal system (with respect to the L2 norm) on a domain $D$ (for e.g., 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 (say, the L2 distance between the boundary of $D$ and the boundary of $D'$ is upper bounded). What can be said about the basis of $D'$ ? Is there any sense in which the functions in the basis of $D$ are "close" to the functions in the basis of $D'$ ? Any known results along these or similar lines appreciated.