A Two Term Kuznecov Sum Formula

Abstract

The Kuznecov sum formula, proved by Zelditch in the Riemannian setting (Zelditch in Comm Part Differ Equ 17(1–2):221–260, 1992), is an asymptotic sum formula

$$\begin{aligned} N(\lambda ):= \sum _{\lambda _j \le \lambda } \left| \int _H e_j \, dV_H \right| ^2 = C_{H,M} \lambda ^{{\text {codim}}H} + O(\lambda ^{{\text {codim}}H - 1}) \end{aligned}$$

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. Assuming that the looping time set is countable, we show for some suitable definition of ‘$\sim$’,

$$\begin{aligned} N(\lambda ) \sim C_{H,M} \lambda ^{{\text {codim}}H} + Q(\lambda ) \lambda ^{{\text {codim}}H - 1} + o(\lambda ^{{\text {codim}}H - 1}) \end{aligned}$$

where Q is a bounded oscillating term and is expressed in terms of the geodesics which depart and arrive H in the normal directions. Our result generalizes a theorem of Safarov on the pointwise Weyl law (Safarov in Funktsional Anal i Prilozhen 22(3):53–65, 1988) in this case. In Canzani et al. (Commun Math Phys 360(2):619–637, 2018) and Canzani and Galkowski (Duke Math J 168(16):2991–3055, 2019) 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 holds if the recurrent directions have 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–Zelditch (Revista Matemática Iberoamericana 32(3):971–994, 2016) and Galkowski (Ann Inst Fourier Grenoble 69(4):1757–1798, 2019).