Asymptotic behavior of $L^2$-normalized eigenfunctions of the Laplace-Beltrami operator on a closed Riemannian manifold

Abstract

Let $e(x,y,ł)$ be the spectral function and $χ_ł$ the unit band spectral projection operator, with respect to the Laplace-Beltrami operator $\D_M$ on a closed Riemannian manifold $M$. We firstly review the one-term asymptotic formula of $e(x,x,ł)$ as $ł\to\infty$ by H{\" o}rmander (1968) and the one of $\p^\al_x\p^\bt_y e(x,y,ł)|_{x=y}$ as $ł\to\infty$ in a geodesic normal coordinate chart by the author (2004) and the sharp asymptotic estimates from above of the mapping norm $\|χ_ł\|_{L_2\to L_p}$ ($2\leq p\leq\infty$) by Sogge (1988 $&$ 1989) and of the mapping norm $\|χ_ł\|_{L_2\to {\rm Sobolev} L_p}$ by the author (2004). In the paper we show the one term asymptotic formula for $e(x,y,ł)$ as $ł\to\infty$, provided that the Riemannian distance between $x$ and $y$ is ${\rm O}(1/ł)$. As a consequence, we obtain the sharp estimate of the mapping norm $\|χ_ł\|_{L_2\to C^\d}$ ($0<\d<1$), where $C^\d(M)$ is the space of H{\" o}lder continuous functions with exponent $\d$ on $M$. Moreover, we show a geometric property of the eigenfunction $e_ł$: $\D_M e_ł+ł^2 e_ł=0$, which says that $1/ł$ is comparable to the distance between the nodal set of $e_ł$ (where $e_ł$ vanishes) and the concentrating set of $e_ł$ (where $e_ł$ attains its maximum or minimum) as $ł\to\infty$.