论文标题
两个任期Kuznecov和公式
A two term Kuznecov sum formula
论文作者
论文摘要
Zelditch在Riemannian设置中证明的Kuznecov和公式是渐近总和公式$$ n（λ）：= \ sum_ {λ_j\leqλ} \ left | \ int_h e_j \，dv_h \ right |^2 = c_ {h，m}λ^{\ propatatOrname {codim} h} + o（λ^{\λ^{\ operatoRatorname {codim} h -1} h -1} h -1} h -1}） $ m $带$δ_ge_j =-λ_j^2 e_j $，而$ h $是嵌入式submanifold。我们为“ $ \ sim $'，$$ n（λ）\ sim c_ {有界的振荡期限，并根据在正常方向出发并到达$ h $的大地测量学表达。 在Canzani，Galkowski和Toth的工作中，它们（作为涉及缺陷度量的更强结果的必然结果），如果正常的地理学重复方向集合为$ h $，则衡量了零，那么我们在SUM中获得了SUM中的个体项的改进界限，即时期积分。我们能够提供动态条件，以使$ Q $是均匀连续的，并且可以用`$ = $'代替``$ \ sim $''。这意味着对周期积分的界限有所改善，并且该条件比测量零的复发方向弱。此外，我们的结果意味着，如果在第一张返回地图下没有$ sn^*h $上的$ l^1 $度量，则需要提高周期积分的界限。这概括了Sogge和Zelditch和Galkowski的定理。
The Kuznecov sum formula, proved by Zelditch in the Riemannian setting, is an asymptotic sum formula $$N(λ) := \sum_{λ_j \leq λ} \left| \int_H e_j \, dV_H \right|^2 = C_{H,M} λ^{\operatorname{codim} H} + O(λ^{\operatorname{codim} H - 1})$$ where $e_j$ constitute a Hilbert basis of Laplace-Beltrami eigenfunctions on a Riemannian manifold $M$ with $Δ_g e_j = -λ_j^2 e_j$, and $H$ is an embedded submanifold. We show for some suitable definition of `$\sim$', $$ N(λ) \sim C_{H,M} λ^{\operatorname{codim} H} + Q(λ) λ^{\operatorname{codim} H - 1} + o(λ^{\operatorname{codim} H - 1}) $$ where $Q$ is a bounded oscillating term and is expressed in terms of the geodesics which depart and arrive $H$ in the normal directions. In work by Canzani, Galkowski, and Toth, they establish (as a corollary to a stronger result involving defect measures) that if the set of recurrent directions of geodesics normal to $H$ has measure zero, then we obtain improved bounds on the individual terms in the sum -- the period integrals. We are able to give a dynamical condition such that $Q$ is uniformly continuous and `$\sim$' can be replaced with `$=$'. This implies improved bounds on period integrals, and this condition is weaker than the recurrent directions having measure zero. Moreover, our result implies improved bounds for period integrals if there is no $L^1$ measure on $SN^*H$ that is invariant under the first return map. This generalizes a theorem of Sogge and Zelditch and of Galkowski.
