Asymptotic behavior of $L^2$-normalized eigenfunctions of the Laplace-Beltrami operator on a closed Riemannian manifold
Abstract
Description
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 $ł\to\infty$.
20 pages, misprints corrected. to appear in the Proceedings of Harmonic Analysis and its Applications at Osaka
20 pages, misprints corrected. to appear in the Proceedings of Harmonic Analysis and its Applications at Osaka