---
title: Kuznecov Sum Formula Overview
url: https://www.emergentmind.com/topics/kuznecov-sum-formula
type: topic
---

# Kuznecov Sum Formula Overview

Searching arXiv for recent papers on the Kuznecov sum formula and its refinements.
The Kuznecov sum formula is an asymptotic statement for spectral sums of squared period integrals of Laplace eigenfunctions over a submanifold, relating high-frequency spectral data to the geometry of conormal directions and, in refined forms, to the dynamics of geodesic returns [2204.13525]. In its classical Riemannian form, for a compact Riemannian manifold without boundary and an embedded compact submanifold \(H\), the formula describes the growth of
\[
N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2
\]
as \(\lambda\to\infty\), where \(\{e_j\}\) is an \(L^2\)-orthonormal Laplace–Beltrami eigenbasis. Subsequent work has identified a structured oscillatory second term governed by conormal loopings [2204.13525], shown that this term is generically absent for a residual set of metrics when the submanifold is fixed [2507.06887], and extended the leading-order theory from smooth submanifold measures to \(s\)-Ahlfors regular fractal measures [2512.18379].

## 1. Classical formulation

In the standard geometric setting, \((M,g)\) is a compact Riemannian manifold without boundary, \(H\subset M\) is an embedded compact submanifold of dimension \(d\), \(n=\dim M\), and \(k=\operatorname{codim}H=n-d\). The Laplace–Beltrami operator \(\Delta_g\) admits an \(L^2(M)\)-orthonormal basis of eigenfunctions \(e_j\) satisfying
\[
\Delta_g e_j=-\lambda_j^2 e_j,
\]
and the period integral over \(H\) is
\[
I_j(H):=\int_H e_j\,dV_H.
\]
The associated Kuznecov counting sum is
\[
N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2.
\]

The classical Kuznecov sum formula, proved by Zelditch in the Riemannian setting, states that there exists a constant \(C_{H,M}\) depending on the geometry of \(M\) and \(H\) such that
\[
N(\lambda)=C_{H,M}\lambda^{\operatorname{codim}H}+O\big(\lambda^{\operatorname{codim}H-1}\big).
\]
In the notation \(k=n-d\), the leading constant is
\[
C_{H,M}=(2\pi)^{-k}\,\operatorname{vol}(H)\,\operatorname{vol}(B^k),
\]
where \(B^k\subset\mathbb{R}^k\) is the unit ball and \(\operatorname{vol}(B^k)=\operatorname{vol}(S^{k-1})/k\) [2204.13525].

For hypersurfaces, \(k=1\) and \(\operatorname{vol}(B^1)=2\), so
\[
C_{H,M}=\frac{1}{\pi}\operatorname{vol}(H).
\]
The formula yields the standard bound
\[
\left|\int_H e_j\,dV_H\right|=O\!\left(\lambda_j^{\frac{\operatorname{codim}H-1}{2}}\right)
=O\!\left(\lambda_j^{\frac{n-d-1}{2}}\right),
\]
and, for every \(\epsilon>0\), a density-\(1\) subsequence satisfies
\[
\left|\int_H e_{j_k}\,dV_H\right|=O\!\big(\lambda_{j_k}^{-\frac d2+\epsilon}\big)
\]
[2204.13525].

A weighted formulation is also standard. For a closed embedded submanifold \(\Sigma\subset M\) of codimension \(k\), induced measure \(d\sigma_g\), and \(a\in C_c^\infty(\Sigma)\), one considers
\[
I_j(\Sigma,a)=\int_\Sigma a(s)\phi_j(s)\,d\sigma_g(s),
\qquad
S(\lambda;\Sigma,a)=\sum_{\lambda_j\le \lambda}|I_j(\Sigma,a)|^2.
\]
The leading coefficient then takes the form
\[
C_0(\Sigma,a,g)=c_{n,k}\int_\Sigma a(s)^2\,d\sigma_g(s),
\qquad
c_{n,k}=(2\pi)^{-k}\omega_{k-1},
\]
equivalently as a fiber integral over the conormal bundle [2507.06887].

## 2. Microlocal and dynamical structure

The refined theory is organized around the geodesic flow on the conormal bundle. Let
\[
p(x,\xi)=|\xi|_{g(x)}
\]
be the principal symbol of \(\sqrt{-\Delta_g}\), and let
\[
G^t:\dot T^*M\to \dot T^*M
\]
be the homogeneous geodesic flow generated by \(H_p\). The punctured conormal bundle and its unit sphere bundle are
\[
\dot N^*H=\{(x,\xi)\in \dot T^*M:x\in H,\ \xi|_{T_xH}=0\},
\qquad
SN^*H=\dot N^*H\cap p^{-1}(1).
\]

For \(t\ne 0\), the relevant return set is the “structured” conormal looping set
\[
\Sigma_t=\{(x,\xi)\in \dot N^*H:G^t(x,\xi)\in \dot N^*H
\ \text{and}\ 
dG^t:T_{(x,\xi)}\dot N^*H\to T_{G^t(x,\xi)}\dot N^*H
\ \text{is a linear isomorphism}\},
\]
with \(S\Sigma_t=\Sigma_t\cap SN^*H\) [2204.13525].

In local coordinates \(x=(x',x'')\) with \(H=\{x''=0\}\), a natural density on \(\dot N^*H\) is
\[
\frac{|g_H(x')|}{|g(x)|^{1/2}}\,|dx'\,d\xi''|,
\]
and the Leray measure on \(SN^*H\) is its restriction to \(p(x,\xi)=1\). If \((y,\eta)=G^t(x,\xi)\), the Jacobian factor \(J_t(x,\xi)\ge 0\) is defined through
\[
\frac{|g_H(y')|}{|g(y)|^{1/2}}\,|dy'\,d\eta''|
\simeq
J_t(x,\xi)\,\frac{|g_H(x')|}{|g(x)|^{1/2}}\,|dx'\,d\xi''|.
\]
This Jacobian measures how conormal volume densities transform under the return map, while the associated Maslov factor \(i^{\sigma_t}\) is determined by the local symplectic geometry along the conormal loop [2204.13525].

The lower-order terms of the Kuznecov expansion arise from nontrivial stationary points of the phase in the time integral of \(R_\Sigma U(t)R_\Sigma^*\), where \(U(t)\) is the wave group and \(R_\Sigma\) is the restriction operator. Geometrically, these are precisely the times \(t\) for which the canonical relation has clean fixed points on \(SN^*\Sigma\), meaning looping directions \((x,\xi)\in SN^*\Sigma\) with \(G^t(x,\xi)\in SN^*\Sigma\) [2507.06887]. Closed geodesics meeting \(\Sigma\) orthogonally and non-periodic looping geodesics can both contribute, depending on clean intersection properties [2507.06887].

## 3. Two-term refinement

A two-term refinement was established by Wyman and Xi. Let \(\mathcal T\subset \mathbb R\setminus\{0\}\) denote the countable set of nonzero times such that \(S\Sigma_t\) has positive measure. Define
\[
q(t):=(2\pi)^{-n+d}\int_{S\Sigma_t} i^{\sigma_t}\sqrt{J_t},
\]
and then the bounded oscillatory function
\[
Q(\lambda):=\sum_{t\in\mathcal T} e^{-it\lambda}\,\frac{q(t)}{-it},
\]
understood as the inverse Fourier transform of the distribution \(\sum_t e^{-it\lambda}q(t)\) [2204.13525].

The asymptotic notation used in the refinement is weaker than literal equality. For monotonically increasing tempered \(f\) and tempered \(g\), the notation
\[
f(\lambda)\sim g(\lambda)+o(\lambda^\alpha)
\]
means there exists a decreasing function \(\epsilon(\lambda)\to 0\) such that
\[
g(\lambda-\epsilon(\lambda))-o(\lambda^\alpha)\le f(\lambda)\le g(\lambda+\epsilon(\lambda))+o(\lambda^\alpha),
\]
with the little-\(o\) depending on \(\epsilon\). If \(g\) is uniformly continuous, then \(f(\lambda)=g(\lambda)+o(\lambda^\alpha)\) [2204.13525].

The main two-term asymptotic is
\[
N(\lambda)\sim C_{H,M}\lambda^{n-d}+Q(\lambda)\lambda^{n-d-1}
+o\!\big(\lambda^{n-d-1}\big)+C,
\]
where the constant \(C\) is harmless except when \(k=1\), accounting for an \(O(1)\) contribution in codimension \(1\) after smoothing and integration [2204.13525]. Under mild dynamical hypotheses, the same structure is expressed in the notation of the later generic-metrics work as
\[
N_\Sigma(\lambda)=C_{\Sigma,M}\lambda^k+Q(\lambda)\lambda^{k-1}+c_{\Sigma,M}+o(\lambda^{k-1}),
\]
with the oscillatory term encoded by the looping locus
\[
\{(x,\xi)\in SN^*\Sigma:G^t(x,\xi)\in SN^*\Sigma\}
\]
[2507.06887].

The geometric meaning of \(Q(\lambda)\) is explicit. The contributions to \(q(t)\) come from unit conormal covectors whose geodesics leave \(H\) in the normal direction and return conormally after time \(t\), with \(dG^t\) mapping \(T\dot N^*H\) isomorphically onto \(T\dot N^*H\). The phase \(e^{-it\lambda}\) records the travel time, \(\sqrt{J_t}\) is the return-map Jacobian amplitude, and \(i^{\sigma_t}\) is the Maslov contribution [2204.13525].

A further structural statement is that \(Q\) is bounded, and the function
\[
(n-d)C_{H,M}\lambda+Q(\lambda)
\]
is monotone increasing in \(\lambda\) [2204.13525]. The vanishing criterion is also sharp in the sense proved there:
\[
Q\equiv 0
\quad\text{if and only if}\quad
S\Sigma_t\ \text{has measure zero in }SN^*H\text{ for every }t\in\mathcal T.
\]
In that case,
\[
N(\lambda)=C_{H,M}\lambda^{n-d}+o\!\big(\lambda^{n-d-1}\big)+C
\]
[2204.13525].

## 4. Dynamical hypotheses and consequences for period integrals

The refined formula connects remainder structure to recurrence of conormal geodesics. Let \(\mathcal R\subset SN^*H\) denote the set of recurrent directions, meaning those \((x,\xi)\) such that for every neighborhood \(U\) of \((x,\xi)\), there exists \(t\ne 0\) with \(G^t(x,\xi)\in U\). If \(|\mathcal R|=0\) in \(SN^*H\), then the averaged size of the dynamical coefficients vanishes:
\[
\lim_{T\to\infty}\frac1T\sum_{t\in\mathcal T\cap[-T,T]}|q(t)|=0.
\]
Under this averaging condition, \(Q\) is uniformly continuous, and the asymptotic upgrades from “\(\sim\)” to exact equality:
\[
N(\lambda)=C_{H,M}\lambda^{n-d}+Q(\lambda)\lambda^{n-d-1}
+o\!\big(\lambda^{n-d-1}\big)+C
\]
[2204.13525].

This condition is stated to be weaker than requiring that the recurrent directions have measure zero, because it encodes averaged control over normal-return dynamics rather than a pointwise measure condition [2204.13525]. A second sufficient criterion is formulated in terms of the first return map on \(SN^*H\). If the only invariant \(L^1\) measure for the first return map is the trivial one, then the same averaging condition holds, hence \(Q\) is uniformly continuous and the exact two-term formula follows [2204.13525]. In the self-focal point case \(H=\{x\}\) with a common return time \(T_0\), this recovers the absence-of-invariant-density condition appearing in work of Sogge–Zelditch and Galkowski, now extended from pointwise Weyl laws to period integrals over general submanifolds [2204.13525].

These asymptotics imply improved estimates for individual periods in shrinking spectral windows. If the averaging condition holds and \(\psi_\lambda\) is a normalized quasimode supported in \([\lambda,\lambda+\epsilon(\lambda)]\) with \(\epsilon(\lambda)\searrow 0\), then
\[
\left|\int_H \psi_\lambda\,dV_H\right|
=o\!\left(\lambda^{\frac{n-d-1}{2}}\right),
\]
improving the standard \(O\big(\lambda^{\frac{n-d-1}{2}}\big)\) bound to a little-\(o\) estimate [2204.13525]. In the weighted setting, earlier work cited in the generic-metrics paper states that when the set of looping directions \(\mathcal L(\Sigma)\) has measure zero, one obtains
\[
\int_\Sigma a\,\phi_j\,d\sigma_g=o\!\left(\lambda_j^{(k-1)/2}\right),
\]
and the generic transversality result implies this condition for generic metrics [2507.06887].

## 5. Generic metrics and elimination of the oscillatory term

A later result studies the dependence of the Kuznecov remainder on the metric. Let \(\mathcal G\) be the Fréchet space of \(C^\infty\) Riemannian metrics on a fixed compact manifold \(M\), equipped with the \(C^\infty\) topology, so that “generic” means residual in this Baire space. For a fixed closed embedded submanifold \(\Sigma\subset M\), the main theorem states that there exists a residual set of metrics \(g\in\mathcal G\) such that
\[
\exp\!\left(\tfrac12 H_{p_g}\right):\dot N^*\Sigma\to T^*M
\]
is transversal to \(\dot N^*\Sigma\) at every intersection point [2507.06887].

The corollary is that, for a residual set of metrics \(g\in\mathcal G\),
\[
N_\Sigma(\lambda)=C_{\Sigma,M}\lambda^k+c_{\Sigma,M}+o(\lambda^{k-1}),
\]
and equivalently,
\[
S(\lambda;\Sigma,a)=C_0(\Sigma,a,g)\lambda^k+o(\lambda^{k-1}).
\]
Thus the oscillatory term \(Q(\lambda)\lambda^{k-1}\) is generically eliminated, improving the classical remainder \(O(\lambda^{k-1})\) to \(o(\lambda^{k-1})\) [2507.06887].

The mechanism is geometric and microlocal. The oscillatory second term arises from nonzero times \(t\) for which the canonical relation of \(R_\Sigma U(t)R_\Sigma^*\) has clean fixed points on \(\dot N^*\Sigma\) or \(SN^*\Sigma\). By stationary phase, each clean intersection contributes a term of size \(\sim \lambda^{k-1}\) with phase \(e^{-it\lambda}\). The generic transversality theorem implies that \(\exp(\tfrac12 H_{p_g})(\dot N^*\Sigma)\cap \dot N^*\Sigma\) is transverse; since both manifolds are half-dimensional in \(T^*M\), the intersections are isolated and countable as \(t\) varies. Consequently, the set of looping directions \(\mathcal L(\Sigma)\) is countable, hence measure zero in \(SN^*\Sigma\), and the measure-zero elimination theorem from the two-term analysis applies [2507.06887].

The proof uses a two-step perturbative strategy. First, localized diffeomorphism-induced perturbations move the embedding of \(\Sigma\) while preserving geodesics up to conjugacy in \(T^*M\), removing periodic geodesics meeting \(\Sigma\) orthogonally. Second, localized conformal perturbations supported near terminal segments of non-closed geodesic arcs are constructed in Fermi-normal coordinates to force surjectivity of the relevant Jacobian and hence transversality. A parametric transversality lemma is then used to obtain a residual set of metrics with the desired property [2507.06887].

A limitation stated there is that one cannot expect simultaneous elimination of oscillatory terms for all submanifolds \(\Sigma\) under a single metric; small geodesic spheres typically produce nontrivial oscillatory terms \(Q(\lambda)\) [2507.06887]. This suggests that generic vanishing is fundamentally a statement for a fixed submanifold rather than a uniform statement over all possible restrictions.

## 6. Variants, examples, and extensions to singular measures

Several model cases clarify the role of conormal returns. For the flat torus \(M=\mathbb T^2\) and \(H\) the unit circle, one computes
\[
N(\lambda)=\sum_{|m|\le \lambda}|J_0(|m|)|^2=2\lambda-\cos(2\lambda)+o(1)+C.
\]
Here the oscillatory term corresponds to diametral normal geodesics of the circle with return time \(|t|=2\); \(S\Sigma_{\pm 2}\) has positive measure, while \(S\Sigma_t\) has measure zero for \(t\neq 0,\pm 2\) [2204.13525]. For a triaxial ellipsoid of dimension \(2\), with \(H\) a geodesic circle equidistant from an umbilical point and its antipode, all geodesics normal to \(H\) pass through the umbilical point and return with periods that are integer multiples of a base time \(T_0/2\). In that case \(S\Sigma_t=SN^*H\) when \(t\in \frac{T_0}{2}\mathbb Z\) and is empty otherwise, while recurrent directions have measure zero, so the exact two-term asymptotic applies [2204.13525].

The framework has also been extended beyond smooth submanifold measures. For a finite Borel measure \(\mu\) on a compact connected Riemannian manifold, one defines
\[
N_\mu(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_M e_j\,d\mu\right|^2.
\]
If \(\mu\) is \(s\)-Ahlfors regular for some \(s\in(0,n)\) and admits an averaged \(s\)-density constant \(A_\mu\), then
\[
N_\mu(\lambda)=(2\pi)^{-(n-s)}\,\operatorname{vol}(B^{n-s})\,A_\mu\,\lambda^{n-s}
+o(\lambda^{n-s})
\qquad (\lambda\to\infty)
\]
[2512.18379].

This extends the classical smooth-submanifold Kuznecov formula by replacing the dimension \(k\) of a smooth submanifold measure with the Ahlfors dimension \(s\) of \(\mu\). In the smooth case, taking \(\mu=d\sigma_Y\) for an embedded \(k\)-dimensional submanifold \(Y\subset M\), one recovers
\[
N_{\sigma_Y}(\lambda)=(2\pi)^{-(n-k)}\operatorname{vol}(B^{n-k})\operatorname{vol}(Y)\lambda^{n-k}+o(\lambda^{n-k}),
\]
matching the leading term of Zelditch’s formula, while the two-term refinement remains specific to the smooth setting [2512.18379]. The point mass case \(\mu=\delta_x\) reduces formally to the local Weyl law, and the result also applies to self-similar or self-conformal measures satisfying the stated density hypotheses [2512.18379].

The fractal generalization is based on heat-kernel regularization:
\[
H_\mu(t):=\sum_{j=0}^\infty e^{-t\lambda_j^2}\left|\int_M e_j\,d\mu\right|^2
=\iint_{M\times M} p_t(x,y)\,d\mu(x)\,d\mu(y),
\]
followed by a Tauberian passage from small-\(t\) asymptotics to spectral asymptotics. The result is a one-term formula only: the remainder is \(o(\lambda^{n-s})\), and in general this cannot be improved uniformly to a power-saving bound for all \(s\)-Ahlfors regular measures admitting an averaged \(s\)-density [2512.18379]. A plausible implication is that the oscillatory second-term mechanism of the smooth theory depends on microlocal structures, such as conormal canonical relations and Maslov data, that are unavailable in the same form for general singular measures.

## 7. Methods, interpretation, and open directions

The proof of the two-term formula proceeds through the Fourier transform of \(N'(\lambda)\). One writes
\[
\widehat{N'}(t)\,|dt|^{1/2}=\big(U(\delta_H)(t),\delta_H\big),
\]
where \(U(t,x,y)=e^{-it\sqrt{-\Delta_g}}(x,y)\) is the half-wave kernel and \(\delta_H\) is the restriction distribution. The Hadamard parametrix for \(\cos(t\sqrt{-\Delta_g})\), together with oscillatory integral representations, is used to analyze singularities at \(t=0\) and \(t\neq 0\) [2204.13525].

At \(t=0\), the principal singularity produces the main term \(C_{H,M}\lambda^{n-d}\). A stationary phase lemma adapted to real oscillatory integrals with real symbols shows that the subprincipal part at \(t=0\) vanishes, permitting precise computation of \(N'*\rho(\lambda)\) to order \(\lambda^{n-d-1}\) before integration in \(\lambda\) [2204.13525]. Away from \(t=0\), singularities arise from geodesics that leave and return conormally to \(H\); these yield the oscillatory contribution of size \(\lambda^{n-d-1}\) with phase \(e^{-it\lambda}\), amplitude determined by \(J_t\), and Maslov factor \(i^{\sigma_t}\) [2204.13525].

A refined Tauberian theorem, identified there as Safarov’s Theorem B.4.1, is used to pass from smoothed quantities to asymptotics of \(N(\lambda)\) [2204.13525]. Global clean composition is not assumed; instead, the analysis uses delicate oscillatory integral estimates, including a “very stationary phase” lemma handling flat-phase situations and a local normal form near conormal return points satisfying the structured-looping condition [2204.13525]. In the generic-metrics work, the same wave-kernel and stationary-phase perspective is combined with transversality theory to show that nonzero-time stationary points are generically negligible [2507.06887].

Several restrictions remain explicit in the literature summarized here. The smooth two-term formula is proved for compact manifolds without boundary and smooth embedded compact submanifolds; extensions to manifolds with boundary would require addressing boundary wave propagation and related parametrix issues [2204.13525]. The structured looping condition and the Maslov/Jacobian data are microlocal and rely on smooth geodesic dynamics on \(SN^*H\) [2204.13525]. In codimension \(1\), the constant \(C\) or \(c_{\Sigma,M}\) appears due to smoothing and integration effects; in all known examples, the later paper states that \(c_{\Sigma,M}=0\) [2507.06887].

Open questions identified in the smooth theory include characterizing dynamical regimes ensuring the averaging condition, computing \(Q(\lambda)\) explicitly in additional model geometries, strengthening individual period bounds under weaker assumptions, and extending the theory to manifolds with boundary and to other spectral quantities involving restriction to \(H\) [2204.13525]. In the generic-metrics setting, the central conclusion is that the leading Kuznecov term is universal and local, while the second term is a dynamical signature of clean conormal loopings; for a Baire-generic class of metrics with fixed \(\Sigma\), such loopings are destroyed by transversality and the oscillatory second term vanishes [2507.06887].

Source: https://www.emergentmind.com/topics/kuznecov-sum-formula