A two term Kuznecov sum formula

TL;DR

This paper refines the Kuznecov sum formula by identifying conditions under which the oscillating term can be precisely characterized, leading to improved bounds on period integrals in spectral geometry.

Contribution

It introduces a dynamical condition that ensures the oscillating term is well-behaved, allowing for exact asymptotics and improved bounds on period integrals beyond previous measure-zero assumptions.

Findings

Established conditions for the oscillating term Q to be uniformly continuous.

Derived improved bounds on period integrals under weaker dynamical assumptions.

Generalized previous results by Sogge, Zelditch, Galkowski regarding invariant measures and recurrence.

Abstract

The Kuznecov sum formula, proved by Zelditch in the Riemannian setting, is an asymptotic sum formula $$N(\lambda) := \sum_{\lambda_j \leq \lambda} \left| \int_H e_j \, dV_H \right|^2 = C_{H,M} \lambda^{\operatorname{codim} H} + O(\lambda^{\operatorname{codim} H - 1})$$ where $e_j$ constitute a Hilbert basis of Laplace-Beltrami eigenfunctions on a Riemannian manifold $M$ with $\Delta_g e_j = -\lambda_j^2 e_j$, and $H$ is an embedded submanifold. We show for some suitable definition of `$\sim$', $$ N(\lambda) \sim C_{H,M} \lambda^{\operatorname{codim} H} + Q(\lambda) \lambda^{\operatorname{codim} H - 1} + o(\lambda^{\operatorname{codim} H - 1}) $$ 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.