Let $(S^2, g)$ be a Riemannian sphere and let $L := \Delta_{S^2} + q$ be a Schrödinger operator on $S^2$. Suppose that $L$ has index equal to one and that $u \in C^{\infty}(S^2)$ ($u \neq 0$) lies in the kernel of $L$. Since $u$ changes sign, it has exactly $2$ nodal domains, and Theorem 2.5 by Shiu-Yuen Cheng in "Eigenfunctions and Nodal Sets" shows that the nodal set $$C = \{ x \in S^2 : u(x) = 0 \}$$ must be a $C^2$ immersed circle. Can we expect that $C$ is in fact a closed geodesic of $(S^2,g)$?
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