부산대학교 도서관

Let $(M,g)$ be a closed, oriented Riemannian manifold, and let $H_\Lambda$ be the subspace of $L^2(M)$ that is spanned by the eigenfunctions of the Laplacian on $M$ that correspond to the eigenvalues $\leq N^2$. Let $N(\Lambda)$ be the dimension of $H_\Lambda$, and let $\mu_\Lambda$ be the Gaussian measure on $H_\Lambda$ with the standard deviation $1/\sqrt{N(\Lambda)}$. The nodal set of almost every (with respect to the measure $\mu_\Lambda$) function $f\in H_\Lambda$ is a smooth hyper-surface in $M$; for a compactly supported $n$-form on $T^*M$, $\langle N^*(f=0),\omega\rangle$ is the integral of $\omega$ over the conormal bundle of the nodal set of $f$. The main result of the paper under review is that $$ \int_{H_\Lambda}\langle N^*(f=0),\omega\rangle d\mu_\Lambda=C_n\Lambda^n\int_{T*M}\pi^*\Omega_g\wedge \omega+O(\Lambda^{n-1}), $$ where $\pi^*\Omega_g$ is the lift to $T^*M$ of the volume form on $M$ and $C_n$ is an explicit constant; $C_n\ne 0$ for odd values of $n$ and $C_n=0$ for even values of $n$. The proof uses the technique of Berezin integrals.