Recently I considered the following question: If we give a second eigenfuntions $g$ on sphere, then can we construct a first eigenfuntions $f$ by $g$? Is there any relationship between the first eigenfuntions and the second eigenfuntions on sphere?
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1$\begingroup$ Eigenfunctions of what? I propose a theorem: Given two arbitrary functions, they are first and second eigenfunctions of something. $\endgroup$– Michael RenardyCommented Apr 23, 2021 at 3:18
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$\begingroup$ Eigenfunctions of Laplacian Operator on sphere $\endgroup$– 管山林Commented Apr 23, 2021 at 3:20
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2$\begingroup$ Isn't the first eigenfunction of the Laplacian on the sphere simply a constant? What am I misunderstanding here? $\endgroup$– Michael RenardyCommented Apr 23, 2021 at 3:27
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$\begingroup$ It is trivial , the first eigenvalue is not zero in our general asumption. $\endgroup$– 管山林Commented Apr 23, 2021 at 3:50
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Eigenfunctions of the Laplace operator for the round curvature 1 metric on the sphere $S^n(1) \subset \mathbb{R}^{n+1}$ are restrictions of homogeneous harmonic polynomials in $\mathbb{R}^{n+1}$. The restriction of a homogeneous degree $d$ harmonic polynomial gives you an eigenfunction for $\Delta_{S^{n-1}(1)}$ with eigenvalue $d(d+n-2)$. These eigenfunctions are well studied and are known as spherical harmonics, they appear in the representation theory for $SO(n+1)$ and many other places, see https://en.wikipedia.org/wiki/Spherical_harmonics
