Kuznecov Sum Formula Overview

• The Kuznecov Sum Formula is an asymptotic statement that relates spectral sums of squared eigenfunction period integrals to the geometric and dynamical properties of submanifolds.
• It utilizes microlocal analysis and stationary phase methods to identify leading and oscillatory terms in the asymptotic expansion of eigenvalue counts.
• Extensions of the formula address singular measures and fractal settings, with generic metric results eliminating the oscillatory second term under transversality conditions.

Searching arXiv for 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 (Wyman et al., 2022). 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 (Wyman et al., 2022), shown that this term is generically absent for a residual set of metrics when the submanifold is fixed (Kaloshin et al., 9 Jul 2025), and extended the leading-order theory from smooth submanifold measures to $s$-Ahlfors regular fractal measures (Xi, 20 Dec 2025).

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 $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$0. The Laplace–Beltrami operator $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$1 admits an $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$2-orthonormal basis of eigenfunctions $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$3 satisfying

$$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$4

and the period integral over $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$5 is

$$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$6

The associated Kuznecov counting sum is

$$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$7

The classical Kuznecov sum formula, proved by Zelditch in the Riemannian setting, states that there exists a constant $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$8 depending on the geometry of $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$9 and $\lambda\to\infty$0 such that

$\lambda\to\infty$1

In the notation $\lambda\to\infty$2, the leading constant is

$\lambda\to\infty$3

where $\lambda\to\infty$4 is the unit ball and $\lambda\to\infty$5 (Wyman et al., 2022).

For hypersurfaces, $\lambda\to\infty$6 and $\lambda\to\infty$7, so

$\lambda\to\infty$8

The formula yields the standard bound

$\lambda\to\infty$9

and, for every $\{e_j\}$0, a density-$\{e_j\}$1 subsequence satisfies

$\{e_j\}$2

(Wyman et al., 2022).

A weighted formulation is also standard. For a closed embedded submanifold $\{e_j\}$3 of codimension $\{e_j\}$4, induced measure $\{e_j\}$5, and $\{e_j\}$6, one considers

$\{e_j\}$7

The leading coefficient then takes the form

$\{e_j\}$8

equivalently as a fiber integral over the conormal bundle (Kaloshin et al., 9 Jul 2025).

2. Microlocal and dynamical structure

The refined theory is organized around the geodesic flow on the conormal bundle. Let

$\{e_j\}$9

be the principal symbol of $L^2$0, and let

$L^2$1

be the homogeneous geodesic flow generated by $L^2$2. The punctured conormal bundle and its unit sphere bundle are

$L^2$3

For $L^2$4, the relevant return set is the “structured” conormal looping set

$L^2$5

with $L^2$6 (Wyman et al., 2022).

In local coordinates $L^2$7 with $L^2$8, a natural density on $L^2$9 is

$s$0

and the Leray measure on $s$1 is its restriction to $s$2. If $s$3, the Jacobian factor $s$4 is defined through

$s$5

This Jacobian measures how conormal volume densities transform under the return map, while the associated Maslov factor $s$6 is determined by the local symplectic geometry along the conormal loop (Wyman et al., 2022).

The lower-order terms of the Kuznecov expansion arise from nontrivial stationary points of the phase in the time integral of $s$7, where $s$8 is the wave group and $s$9 is the restriction operator. Geometrically, these are precisely the times $(M,g)$0 for which the canonical relation has clean fixed points on $(M,g)$1, meaning looping directions $(M,g)$2 with $(M,g)$3 (Kaloshin et al., 9 Jul 2025). Closed geodesics meeting $(M,g)$4 orthogonally and non-periodic looping geodesics can both contribute, depending on clean intersection properties (Kaloshin et al., 9 Jul 2025).

3. Two-term refinement

A two-term refinement was established by Wyman and Xi. Let $(M,g)$5 denote the countable set of nonzero times such that $(M,g)$6 has positive measure. Define

$(M,g)$7

and then the bounded oscillatory function

$(M,g)$8

understood as the inverse Fourier transform of the distribution $(M,g)$9 (Wyman et al., 2022).

The asymptotic notation used in the refinement is weaker than literal equality. For monotonically increasing tempered $H\subset M$0 and tempered $H\subset M$1, the notation

$H\subset M$2

means there exists a decreasing function $H\subset M$3 such that

$H\subset M$4

with the little-$H\subset M$5 depending on $H\subset M$6. If $H\subset M$7 is uniformly continuous, then $H\subset M$8 (Wyman et al., 2022).

The main two-term asymptotic is

$H\subset M$9

where the constant $d$0 is harmless except when $d$1, accounting for an $d$2 contribution in codimension $d$3 after smoothing and integration (Wyman et al., 2022). Under mild dynamical hypotheses, the same structure is expressed in the notation of the later generic-metrics work as

$d$4

with the oscillatory term encoded by the looping locus

$d$5

(Kaloshin et al., 9 Jul 2025).

The geometric meaning of $d$6 is explicit. The contributions to $d$7 come from unit conormal covectors whose geodesics leave $d$8 in the normal direction and return conormally after time $d$9, with $n=\dim M$0 mapping $n=\dim M$1 isomorphically onto $n=\dim M$2. The phase $n=\dim M$3 records the travel time, $n=\dim M$4 is the return-map Jacobian amplitude, and $n=\dim M$5 is the Maslov contribution (Wyman et al., 2022).

A further structural statement is that $n=\dim M$6 is bounded, and the function

$n=\dim M$7

is monotone increasing in $n=\dim M$8 (Wyman et al., 2022). The vanishing criterion is also sharp in the sense proved there: $n=\dim M$9 In that case,

$$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$00

(Wyman et al., 2022).

4. Dynamical hypotheses and consequences for period integrals

The refined formula connects remainder structure to recurrence of conormal geodesics. Let $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$01 denote the set of recurrent directions, meaning those $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$02 such that for every neighborhood $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$03 of $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$04, there exists $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$05 with $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$06. If $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$07 in $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$08, then the averaged size of the dynamical coefficients vanishes: $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$09 Under this averaging condition, $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$10 is uniformly continuous, and the asymptotic upgrades from “$$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$11” to exact equality: $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$12 (Wyman et al., 2022).

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 (Wyman et al., 2022). A second sufficient criterion is formulated in terms of the first return map on $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$13. If the only invariant $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$14 measure for the first return map is the trivial one, then the same averaging condition holds, hence $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$15 is uniformly continuous and the exact two-term formula follows (Wyman et al., 2022). In the self-focal point case $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$16 with a common return time $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$17, 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 (Wyman et al., 2022).

These asymptotics imply improved estimates for individual periods in shrinking spectral windows. If the averaging condition holds and $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$18 is a normalized quasimode supported in $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$19 with $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$20, then

$$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$21

improving the standard $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$22 bound to a little-$$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$23 estimate (Wyman et al., 2022). In the weighted setting, earlier work cited in the generic-metrics paper states that when the set of looping directions $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$24 has measure zero, one obtains

$$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$25

and the generic transversality result implies this condition for generic metrics (Kaloshin et al., 9 Jul 2025).

5. Generic metrics and elimination of the oscillatory term

A later result studies the dependence of the Kuznecov remainder on the metric. Let $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$26 be the Fréchet space of $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$27 Riemannian metrics on a fixed compact manifold $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$28, equipped with the $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$29 topology, so that “generic” means residual in this Baire space. For a fixed closed embedded submanifold $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$30, the main theorem states that there exists a residual set of metrics $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$31 such that

$$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$32

is transversal to $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$33 at every intersection point (Kaloshin et al., 9 Jul 2025).

The corollary is that, for a residual set of metrics $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$34,

$$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$35

and equivalently,

$$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$36

Thus the oscillatory term $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$37 is generically eliminated, improving the classical remainder $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$38 to $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$39 (Kaloshin et al., 9 Jul 2025).

The mechanism is geometric and microlocal. The oscillatory second term arises from nonzero times $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$40 for which the canonical relation of $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$41 has clean fixed points on $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$42 or $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$43. By stationary phase, each clean intersection contributes a term of size $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$44 with phase $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$45. The generic transversality theorem implies that $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$46 is transverse; since both manifolds are half-dimensional in $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$47, the intersections are isolated and countable as $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$48 varies. Consequently, the set of looping directions $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$49 is countable, hence measure zero in $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$50, and the measure-zero elimination theorem from the two-term analysis applies (Kaloshin et al., 9 Jul 2025).

The proof uses a two-step perturbative strategy. First, localized diffeomorphism-induced perturbations move the embedding of $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$51 while preserving geodesics up to conjugacy in $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$52, removing periodic geodesics meeting $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$53 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 (Kaloshin et al., 9 Jul 2025).

A limitation stated there is that one cannot expect simultaneous elimination of oscillatory terms for all submanifolds $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$54 under a single metric; small geodesic spheres typically produce nontrivial oscillatory terms $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$55 (Kaloshin et al., 9 Jul 2025). 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 $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$56 and $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$57 the unit circle, one computes

$$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$58

Here the oscillatory term corresponds to diametral normal geodesics of the circle with return time $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$59; $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$60 has positive measure, while $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$61 has measure zero for $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$62 (Wyman et al., 2022). For a triaxial ellipsoid of dimension $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$63, with $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$64 a geodesic circle equidistant from an umbilical point and its antipode, all geodesics normal to $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$65 pass through the umbilical point and return with periods that are integer multiples of a base time $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$66. In that case $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$67 when $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$68 and is empty otherwise, while recurrent directions have measure zero, so the exact two-term asymptotic applies (Wyman et al., 2022).

The framework has also been extended beyond smooth submanifold measures. For a finite Borel measure $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$69 on a compact connected Riemannian manifold, one defines

$$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$70

If $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$71 is $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$72-Ahlfors regular for some $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$73 and admits an averaged $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$74-density constant $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$75, then

$$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$76

(Xi, 20 Dec 2025).

This extends the classical smooth-submanifold Kuznecov formula by replacing the dimension $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$77 of a smooth submanifold measure with the Ahlfors dimension $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$78 of $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$79. In the smooth case, taking $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$80 for an embedded $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$81-dimensional submanifold $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$82, one recovers

$$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$83

matching the leading term of Zelditch’s formula, while the two-term refinement remains specific to the smooth setting (Xi, 20 Dec 2025). The point mass case $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$84 reduces formally to the local Weyl law, and the result also applies to self-similar or self-conformal measures satisfying the stated density hypotheses (Xi, 20 Dec 2025).

The fractal generalization is based on heat-kernel regularization: $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$85 followed by a Tauberian passage from small-$$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$86 asymptotics to spectral asymptotics. The result is a one-term formula only: the remainder is $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$87, and in general this cannot be improved uniformly to a power-saving bound for all $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$88-Ahlfors regular measures admitting an averaged $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$89-density (Xi, 20 Dec 2025). 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 $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$90. One writes

$$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$91

where $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$92 is the half-wave kernel and $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$93 is the restriction distribution. The Hadamard parametrix for $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$94, together with oscillatory integral representations, is used to analyze singularities at $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$95 and $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$96 (Wyman et al., 2022).

At $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$97, the principal singularity produces the main term $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$98. A stationary phase lemma adapted to real oscillatory integrals with real symbols shows that the subprincipal part at $$N(\lambda):=\sum_{\lambda_j\le \lambda}\left|\int_H e_j\,dV_H\right|^2$$99 vanishes, permitting precise computation of $\lambda\to\infty$00 to order $\lambda\to\infty$01 before integration in $\lambda\to\infty$02 (Wyman et al., 2022). Away from $\lambda\to\infty$03, singularities arise from geodesics that leave and return conormally to $\lambda\to\infty$04; these yield the oscillatory contribution of size $\lambda\to\infty$05 with phase $\lambda\to\infty$06, amplitude determined by $\lambda\to\infty$07, and Maslov factor $\lambda\to\infty$08 (Wyman et al., 2022).

A refined Tauberian theorem, identified there as Safarov’s Theorem B.4.1, is used to pass from smoothed quantities to asymptotics of $\lambda\to\infty$09 (Wyman et al., 2022). 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 (Wyman et al., 2022). 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 (Kaloshin et al., 9 Jul 2025).

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 (Wyman et al., 2022). The structured looping condition and the Maslov/Jacobian data are microlocal and rely on smooth geodesic dynamics on $\lambda\to\infty$10 (Wyman et al., 2022). In codimension $\lambda\to\infty$11, the constant $\lambda\to\infty$12 or $\lambda\to\infty$13 appears due to smoothing and integration effects; in all known examples, the later paper states that $\lambda\to\infty$14 (Kaloshin et al., 9 Jul 2025).

Open questions identified in the smooth theory include characterizing dynamical regimes ensuring the averaging condition, computing $\lambda\to\infty$15 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 $\lambda\to\infty$16 (Wyman et al., 2022). 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 $\lambda\to\infty$17, such loopings are destroyed by transversality and the oscillatory second term vanishes (Kaloshin et al., 9 Jul 2025).