Intersection bounds for nodal sets of Laplace eigenfunctions (Q2419133)

Let $(M,g)$ be a closed real analytic Riemannian manifold. Let $\phi_\lambda$ be an eigenfunction of the associated Laplace operator corresponding to the eigenvalue $\lambda^2$ normalized so $\|\phi_\lambda\|_{L^2}=1$. The authors study the nodal set (i.e., the zero set) $Z_{\phi_\lambda}:=\phi_\lambda^{-1}\{0\}$ of $\phi_\lambda$ as $\lambda$ tends to infinity. Let $H$ be a fixed real analytic closed curve in $M$. \par The authors study the local structure of $Z_{\phi_\lambda}$ by examining $n(H,\phi_\lambda):=|Z_{\phi_\lambda}\cap H|$ which is the number of points in the intersection of $Z_{\phi_\lambda}$ with $H$. One has to place conditions on the curve $H$. For example, if $(M,g)$ is the round sphere and $H$ is the equator, then $n(H,\phi_\lambda)$ is infinite for any odd spherical harmonic. A curve is said to be \textit{good} if for $\lambda$ sufficiently large, $\|\phi_\lambda\|_{L^2}(H)\ge e^{-C\lambda}$. The authors generalize previous results for domains in the plane to show \par Theorem 1. If $H$ is a good curve, then $n(H,\phi_\lambda)=O(\lambda)$. \par There is a related concept called \textit{weak goodness}; a curve $H$ is said to be \textit{weakly good} if $\sup_{z\in H^{\mathbb{C}}}|\phi_\lambda^{\mathbb{C}}(z)|\ge e^{-c\lambda}$ for some $c>0$ where the sup is taken over a Grauert subtube in the complexification. The paper contains a clear historical discussion of the problem. Section 1 deals with intersection bounds on real analytic spaces and establishes Theorem 1. Section 2 shows that weak goodness and goodness are equivalent in the setting at hand. For the entire collection see [Zbl 1412.32002].