Microlocal Weyl Laws: Asymptotic Spectral Analysis

• Microlocal Weyl laws are asymptotic formulas that refine Weyl’s law by accounting for phase-space localization using Fourier integral operator parametrices and stationary phase analysis.
• They offer a detailed methodology to extract local spectral projector asymptotics by microlocalizing the principal symbol and applying Tauberian arguments.
• These laws have wide-ranging applications, including symmetry reduction on Lie groups, pointwise eigenvalue counts, and adaptations to non-selfadjoint or subelliptic operator settings.

Searching arXiv for recent and foundational papers on microlocal Weyl laws. Microlocal Weyl laws are asymptotic formulas that refine Weyl’s law by resolving spectral data in phase space rather than only at the level of a global counting function. In their standard form, they describe the high-frequency or semiclassical behavior of spectral projector kernels, microlocalized traces, or symmetry-reduced counting functions in terms of Liouville-type phase-space volumes associated with the principal symbol and its Hamiltonian flow. Across the literature, the basic analytic architecture is stable: Fourier integral operator parametrices for propagators, stationary phase analysis of oscillatory integrals, and Tauberian passage from smoothed to sharp spectral quantities (Ivrii, 2016).

1. Classical formulation and microlocal content

The classical global Weyl law counts eigenvalues by phase-space volume. For a self-adjoint elliptic operator of order $m$, the leading term is the volume of $\{(x,\xi): p(x,\xi)\le \lambda\}$, and the local or microlocal form replaces this global phase-space volume by integrals of the principal symbol against microlocal cutoffs. In the semiclassical survey literature, this is expressed through the spectral projector kernel $e(x,y,\lambda)$ and microlocalizations $Q_1,Q_2$, with principal coefficients

$$\kappa_0(\lambda,x)=(2\pi)^{-d}\int q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,d\xi,$$

and

$$\varkappa_0(\lambda)=(2\pi)^{-d}\iint q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,dx\,d\xi,$$

so that local and global Weyl asymptotics are literally symbol-level phase-space integrals (Ivrii, 2016).

This microlocal viewpoint is not confined to scalar pseudodifferential calculus on manifolds. On compact Lie groups, the local Weyl formula can be reformulated in the language of matrix-valued quantisation: for $A\in \Psi^0(G)$,

$$\sum_{[\xi]\in\widehat G:\,|\xi|\le \lambda} d_\xi \int_G \mathrm{Tr}\bigl(\sigma_{\mathrm{glob}}(x,[\xi])\bigr)\,dx = \int_{T^*S(G)} \sigma_{\mathrm{loc}}(x,\eta)\,d\mu_L(x,\eta)\,\lambda^n + O(\lambda^{n-1}),$$

so the spectral side is reorganized representation-theoretically while the leading term remains a Liouville average on the co-sphere bundle (Cardona et al., 2022). A common misconception is that a Weyl law is only a statement about the raw counting function $N(\lambda)$; microlocal Weyl laws are more precise because they retain the phase-space localization, and in some settings they retain off-diagonal kernel structure as well.

2. Propagators, oscillatory integrals, and Tauberian passage

The core microlocal mechanism is the singularity structure of the propagator. In the standard FIO framework, the half-density propagator $U_{1/2}$ has canonical relation

$\{(x,\xi): p(x,\xi)\le \lambda\}$0

where $\{(x,\xi): p(x,\xi)\le \lambda\}$1 is the Hamiltonian flow of the principal symbol $\{(x,\xi): p(x,\xi)\le \lambda\}$2. The singularity of $\{(x,\xi): p(x,\xi)\le \lambda\}$3 at $\{(x,\xi): p(x,\xi)\le \lambda\}$4 produces Weyl asymptotics through stationary phase (Ramacher, 2010).

In symmetry-reduced and localized settings, the same principle survives after additional geometric constraints are imposed. Ramacher’s equivariant analysis reduces the trace near $\{(x,\xi): p(x,\xi)\le \lambda\}$5 to oscillatory integrals

$\{(x,\xi): p(x,\xi)\le \lambda\}$6

whose critical set is singular when the action is not free (Ramacher, 2010). For quantum completely integrable systems, the smoothed joint spectral measure is first put into an oscillatory form by a microlocal normal form, and only then is a Tauberian argument used to recover sharp projector asymptotics with $\{(x,\xi): p(x,\xi)\le \lambda\}$7 remainder (Eswarathasan et al., 2024). In the Toeplitz-Kähler setting, partial Bergman kernels are handled analogously, but the underlying parametrix is the Boutet-de-Monvel–Sjöstrand Szegő kernel rather than the wave group (Zelditch et al., 2018).

These examples show that “microlocal Weyl law” is best understood as a methodologically structured class of asymptotic results: first construct a microlocal or FIO representation, then extract the principal term by stationary phase, and finally pass from smoothed to sharp spectral data by a Tauberian theorem.

3. Symmetry, reduction, and singular group actions

A major branch of microlocal Weyl theory concerns symmetry reduction. If a compact Lie group $\{(x,\xi): p(x,\xi)\le \lambda\}$8 acts effectively and isometrically on a compact manifold $\{(x,\xi): p(x,\xi)\le \lambda\}$9, and $e(x,y,\lambda)$0 is an invariant elliptic classical pseudodifferential operator of order $e(x,y,\lambda)$1, then $e(x,y,\lambda)$2 decomposes into isotypic components, and the relevant counting function is the spectral count along a fixed irreducible representation. The equivariant trace formula expresses the corresponding spectral distribution as

$e(x,y,\lambda)$3

with $e(x,y,\lambda)$4 (Ramacher, 2010).

The geometric object governing the asymptotics is the momentum-map zero set

$e(x,y,\lambda)$5

together with the reduced energy surface $e(x,y,\lambda)$6. In the principal orbit case, the equivariant Weyl law is

$e(x,y,\lambda)$7

where $e(x,y,\lambda)$8 is the principal orbit dimension and $e(x,y,\lambda)$9 is the maximal length of a chain of orbit types (Ramacher, 2010). The leading coefficient is therefore a reduced Liouville volume multiplied by the representation-theoretic factor $Q_1,Q_2$0.

The distinctive difficulty is that $Q_1,Q_2$1 and the critical set

$Q_1,Q_2$2

are singular when the action has non-free orbits. Ramacher resolves this obstruction by monoidal transformations along isotropy bundles, obtaining weak transforms with clean critical manifolds and transversally non-degenerate Hessians, thereby restoring stationary phase (Ramacher, 2010). This is one of the clearest demonstrations that microlocal Weyl laws are not merely asymptotic counting statements; they are also desingularization statements about the relevant phase geometry.

The compact Lie group case furnishes a complementary, representation-theoretic reformulation. There the spectral side of the local Weyl formula is rewritten using the global matrix-valued symbol $Q_1,Q_2$3, and the resulting asymptotic identifies the representation average of $Q_1,Q_2$4 with the Liouville average of the principal symbol on $Q_1,Q_2$5 (Cardona et al., 2022).

4. Pointwise laws, joint spectra, and Toeplitz quantization

Microlocal Weyl laws become genuinely pointwise when the asymptotic object is a spectral projector kernel rather than an integrated trace. For quantum completely integrable systems $Q_1,Q_2$6 with first-order commuting self-adjoint pseudodifferential operators and elliptic $Q_1,Q_2$7, the joint projector kernel onto the box

$Q_1,Q_2$8

admits a microlocal pointwise Weyl law on a conic region where the fiber-rank $Q_1,Q_2$9 condition holds (Eswarathasan et al., 2024). After insertion of a zeroth-order cutoff $$\kappa_0(\lambda,x)=(2\pi)^{-d}\int q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,d\xi,$$0, one obtains

$$\kappa_0(\lambda,x)=(2\pi)^{-d}\int q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,d\xi,$$1

with $$\kappa_0(\lambda,x)=(2\pi)^{-d}\int q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,d\xi,$$2 (Eswarathasan et al., 2024). On the diagonal, the amplitude reduces to $$\kappa_0(\lambda,x)=(2\pi)^{-d}\int q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,d\xi,$$3, so the leading term becomes a microlocal density-of-states integral over the fiber.

This result is a joint-spectrum analogue of Hörmander’s pointwise Weyl law. Its novelty lies in replacing the single Hamiltonian by the joint moment map $$\kappa_0(\lambda,x)=(2\pi)^{-d}\int q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,d\xi,$$4, and the ordinary phase by a generating function $$\kappa_0(\lambda,x)=(2\pi)^{-d}\int q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,d\xi,$$5 arising from the microlocal normal form (Eswarathasan et al., 2024).

In the Kähler-Toeplitz setting, pointwise Weyl laws for partial Bergman kernels take a different but parallel form. For the Toeplitz quantization $$\kappa_0(\lambda,x)=(2\pi)^{-d}\int q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,d\xi,$$6 of a Hamiltonian $$\kappa_0(\lambda,x)=(2\pi)^{-d}\int q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,d\xi,$$7 on a polarized Kähler manifold, the smoothed pointwise spectral sums

$$\kappa_0(\lambda,x)=(2\pi)^{-d}\int q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,d\xi,$$8

have a full asymptotic expansion in terms of periodic orbit data. At periodic points,

$$\kappa_0(\lambda,x)=(2\pi)^{-d}\int q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,d\xi,$$9

where $$\varkappa_0(\lambda)=(2\pi)^{-d}\iint q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,dx\,d\xi,$$0 is determined by the holomorphic block $$\varkappa_0(\lambda)=(2\pi)^{-d}\iint q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,dx\,d\xi,$$1 of the linearized return map and the Hamilton vector component $$\varkappa_0(\lambda)=(2\pi)^{-d}\iint q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,dx\,d\xi,$$2 (Zelditch et al., 2018). Under strong hyperbolicity, a semiclassical Tauberian theorem converts this smoothed expansion into a sharp two-term pointwise Weyl law for spectral windows of width $$\varkappa_0(\lambda)=(2\pi)^{-d}\iint q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,dx\,d\xi,$$3 in the rescaled variable (Zelditch et al., 2018).

Together, these papers show that pointwise microlocal Weyl laws do not merely recover a local density; they can also retain explicit phase information and periodic-orbit contributions.

5. Non-selfadjoint, open, and subelliptic analogues

The phrase “microlocal Weyl law” has been extended beyond the self-adjoint elliptic setting, but the strength of the conclusion varies substantially. In dimension two, for analytic non-selfadjoint perturbations $$\varkappa_0(\lambda)=(2\pi)^{-d}\iint q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,dx\,d\xi,$$4 of self-adjoint semiclassical operators with completely integrable flow, the number of eigenvalues in a complex spectral rectangle is governed by the phase-space volume of a dynamically defined region between Diophantine invariant tori: $$\varkappa_0(\lambda)=(2\pi)^{-d}\iint q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,dx\,d\xi,$$5 so the imaginary spectral location is encoded by long-time averages of the perturbation along the classical flow (Hitrik et al., 2011). This is microlocal in the sense that the spectral band is carved out by invariant tori and normal forms, not by a global scalar energy threshold.

For the semiclassical $$\varkappa_0(\lambda)=(2\pi)^{-d}\iint q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,dx\,d\xi,$$6-operator on the complex torus with exponential weight $$\varkappa_0(\lambda)=(2\pi)^{-d}\iint q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,dx\,d\xi,$$7, the number of exponentially small singular values obeys

$$\varkappa_0(\lambda)=(2\pi)^{-d}\iint q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,dx\,d\xi,$$8

where $$\varkappa_0(\lambda)=(2\pi)^{-d}\iint q_1^0(x,\xi)\,\theta\bigl(\lambda-H^0(x,\xi)\bigr)\,q_2^0(x,\xi)\,dx\,d\xi,$$9 is the unique $A\in \Psi^0(G)$0 solution of a double-obstacle problem. The paper identifies the leading term as the symplectic volume of a subset of the $A\in \Psi^0(G)$1-Lagrangian $A\in \Psi^0(G)$2, and the small-$A\in \Psi^0(G)$3 correction has the turning-point scale $A\in \Psi^0(G)$4 (Hitrik et al., 12 May 2025). This is explicitly described there as a Berezin-Toeplitz type Weyl law in a non-selfadjoint setting.

Open and partially open quantum maps exhibit a different phenomenon: the available results are often upper bounds or concentration laws rather than complete asymptotic expansions. For the quantum open baker’s map, the counting function in an annulus satisfies

$A\in \Psi^0(G)$5

with $A\in \Psi^0(G)$6 the trapped-set dimension, so the spectral exponent is determined by the fractal geometry of the trapped set (Li, 2022). For damped quantum maps on the torus, the normalized spectrum concentrates in annuli determined by Birkhoff averages of $A\in \Psi^0(G)$7, and in the ergodic case almost all eigenvalue moduli concentrate near the geometric mean $A\in \Psi^0(G)$8 (Schenck, 2008). These results are best viewed as microlocal Weyl-type analogues rather than strict local Weyl laws.

Subelliptic analogues also require care in terminology. On compact Heisenberg manifolds, the counting function for the family $A\in \Psi^0(G)$9 satisfies

$$\sum_{[\xi]\in\widehat G:\,|\xi|\le \lambda} d_\xi \int_G \mathrm{Tr}\bigl(\sigma_{\mathrm{glob}}(x,[\xi])\bigr)\,dx = \int_{T^*S(G)} \sigma_{\mathrm{loc}}(x,\eta)\,d\mu_L(x,\eta)\,\lambda^n + O(\lambda^{n-1}),$$0

with the exponent $$\sum_{[\xi]\in\widehat G:\,|\xi|\le \lambda} d_\xi \int_G \mathrm{Tr}\bigl(\sigma_{\mathrm{glob}}(x,[\xi])\bigr)\,dx = \int_{T^*S(G)} \sigma_{\mathrm{loc}}(x,\eta)\,d\mu_L(x,\eta)\,\lambda^n + O(\lambda^{n-1}),$$1 reflecting the Heisenberg homogeneous dimension $$\sum_{[\xi]\in\widehat G:\,|\xi|\le \lambda} d_\xi \int_G \mathrm{Tr}\bigl(\sigma_{\mathrm{glob}}(x,[\xi])\bigr)\,dx = \int_{T^*S(G)} \sigma_{\mathrm{loc}}(x,\eta)\,d\mu_L(x,\eta)\,\lambda^n + O(\lambda^{n-1}),$$2 through $$\sum_{[\xi]\in\widehat G:\,|\xi|\le \lambda} d_\xi \int_G \mathrm{Tr}\bigl(\sigma_{\mathrm{glob}}(x,[\xi])\bigr)\,dx = \int_{T^*S(G)} \sigma_{\mathrm{loc}}(x,\eta)\,d\mu_L(x,\eta)\,\lambda^n + O(\lambda^{n-1}),$$3 (Fan et al., 2021). The paper is explicit that it proves a global Weyl law by spectral decomposition and a Tauberian theorem, not a microlocal projector asymptotic. A plausible implication is that the result captures the correct homogeneous “phase-space” scaling without yet supplying the local microlocal measure.

6. Boundaries, cusps, singularities, and robustness of the principal term

Boundary and noncompact geometry alter both the form of the Weyl law and the microlocal proof. In the standard survey framework, the boundary contributes the familiar second term

$$\sum_{[\xi]\in\widehat G:\,|\xi|\le \lambda} d_\xi \int_G \mathrm{Tr}\bigl(\sigma_{\mathrm{glob}}(x,[\xi])\bigr)\,dx = \int_{T^*S(G)} \sigma_{\mathrm{loc}}(x,\eta)\,d\mu_L(x,\eta)\,\lambda^n + O(\lambda^{n-1}),$$4

for Dirichlet and Neumann problems, and pointwise local laws acquire explicit boundary-layer terms (Ivrii, 2016). On manifolds with exact hyperbolic cusps, the natural counting function is not the pure point count but

$$\sum_{[\xi]\in\widehat G:\,|\xi|\le \lambda} d_\xi \int_G \mathrm{Tr}\bigl(\sigma_{\mathrm{glob}}(x,[\xi])\bigr)\,dx = \int_{T^*S(G)} \sigma_{\mathrm{loc}}(x,\eta)\,d\mu_L(x,\eta)\,\lambda^n + O(\lambda^{n-1}),$$5

where $$\sum_{[\xi]\in\widehat G:\,|\xi|\le \lambda} d_\xi \int_G \mathrm{Tr}\bigl(\sigma_{\mathrm{glob}}(x,[\xi])\bigr)\,dx = \int_{T^*S(G)} \sigma_{\mathrm{loc}}(x,\eta)\,d\mu_L(x,\eta)\,\lambda^n + O(\lambda^{n-1}),$$6 is the scattering phase. Its asymptotic expansion contains the compact-type leading term together with cusp corrections

$$\sum_{[\xi]\in\widehat G:\,|\xi|\le \lambda} d_\xi \int_G \mathrm{Tr}\bigl(\sigma_{\mathrm{glob}}(x,[\xi])\bigr)\,dx = \int_{T^*S(G)} \sigma_{\mathrm{loc}}(x,\eta)\,d\mu_L(x,\eta)\,\lambda^n + O(\lambda^{n-1}),$$7

and the remainder improves from $$\sum_{[\xi]\in\widehat G:\,|\xi|\le \lambda} d_\xi \int_G \mathrm{Tr}\bigl(\sigma_{\mathrm{glob}}(x,[\xi])\bigr)\,dx = \int_{T^*S(G)} \sigma_{\mathrm{loc}}(x,\eta)\,d\mu_L(x,\eta)\,\lambda^n + O(\lambda^{n-1}),$$8 to $$\sum_{[\xi]\in\widehat G:\,|\xi|\le \lambda} d_\xi \int_G \mathrm{Tr}\bigl(\sigma_{\mathrm{glob}}(x,[\xi])\bigr)\,dx = \int_{T^*S(G)} \sigma_{\mathrm{loc}}(x,\eta)\,d\mu_L(x,\eta)\,\lambda^n + O(\lambda^{n-1}),$$9 under the measure-zero periodic geodesic condition, or to $N(\lambda)$0 under no conjugate points (Bonthonneau, 2015). Here the wave trace must be replaced by a $N(\lambda)$1-trace, and the relevant microlocal construction is a hyperbolic Hadamard parametrix on the universal cover.

Compact manifolds with boundary and singular lower-order perturbations provide a different robustness result. For $N(\lambda)$2 with $N(\lambda)$3 and $N(\lambda)$4, the counting function obeys

$N(\lambda)$5

and under the measure-zero periodic geodesic billiard condition, the two-term law is

$N(\lambda)$6

with minus for Dirichlet and plus for Neumann (Huang et al., 2024). The paper’s central point is that the order-zero critically singular potential does not alter either the principal symbol or the boundary coefficient.

A related robustness phenomenon appears for the Anderson Hamiltonian on a two-dimensional compact manifold. There the renormalized operator $N(\lambda)$7, defined by high-order paracontrolled calculus, has pure point spectrum and almost surely satisfies

$N(\lambda)$8

the same leading constant as the Laplace–Beltrami operator (Mouzard, 2020). This is explicitly a global Weyl-type law rather than a microlocal local law, but it shows that even highly singular random perturbations can leave the leading asymptotic unchanged.

7. Scope, distinctions, and current methodological boundaries

The contemporary literature uses “microlocal Weyl law” in several adjacent senses, and the distinction is substantive. Some papers prove local or pointwise spectral projector asymptotics with explicit phases and amplitudes, as in quantum completely integrable systems and partial Bergman kernels (Eswarathasan et al., 2024, Zelditch et al., 2018). Others prove integrated or symmetry-reduced counting laws whose proofs are microlocal but whose final statements are global, as in singular equivariant Weyl laws or cusp manifolds (Ramacher, 2010, Bonthonneau, 2015). Still others provide analogues in non-selfadjoint, open, or subelliptic settings where the principal term survives but the final statement may be only an upper bound, a concentration theorem, or a Tauberian global count (Li, 2022, Fan et al., 2021).

A recurring misconception is that all microlocal Weyl problems reduce to the same stationary phase computation on a smooth critical manifold. The singular equivariant case shows otherwise: the phase geometry can be genuinely singular and may require resolution of singularities before stationary phase is applicable (Ramacher, 2010). Another misconception is that any paper on Weyl quantization or dyadic localization already proves a Weyl law. The anisotropic dyadic microlocal partition developed in “Dyadic microlocal partition for anisotropic metrics and uniform Weyl quantization” is explicit that it does not prove trace asymptotics, local Weyl laws, spectral counting, or local density-of-states estimates; it provides a uniform microlocal toolbox—localized quantization bounds, Moyal truncation with controlled remainders, Cotlar–Stein recombination, and a parametrix—that can be inserted into standard Weyl-law arguments (Vergara, 16 Oct 2025).

This suggests a useful taxonomy. A strict microlocal Weyl law gives asymptotics for a microlocalized spectral projector or its trace. A reduced or equivariant microlocal Weyl law replaces the ambient phase space by a reduced one and may require desingularization. A pointwise law resolves the kernel on or near the diagonal. An analogue transfers the phase-space principle to non-selfadjoint, open, or subelliptic settings, but may stop short of a full local projector expansion. The unifying principle remains the same throughout: spectral asymptotics are controlled by the geometry of the relevant Hamiltonian or reduced phase space, and the main technical work lies in making that phase geometry analytically accessible.