Symplectic Weyl Laws

• Symplectic Weyl Laws are a set of asymptotic results that relate spectral invariants to symplectic volumes and group representations in geometric and algebraic contexts.
• They employ advanced methods including symplectic reduction, monoidal transformations, and stationary phase analysis to derive precise leading and subleading eigenvalue estimates.
• These laws further elucidate the impact of packing stability, non-selfadjoint perturbations, and noncommutative geometry, offering quantitative controls on remainder terms and spectral corrections.

Symplectic Weyl Laws describe the asymptotic distribution of spectral invariants, eigenvalues, or algebraic multiplicities in various geometric and algebraic contexts governed by symplectic structures. Unlike classical Weyl laws, which count eigenvalues via phase-space volumes in the absence of symmetries or additional geometric data, symplectic Weyl laws integrate reduction techniques, Hamiltonian flows, group actions, noncommutativity, and quantization procedures to account for symplectic geometry's subtleties and complexities. These laws establish connections between spectral asymptotics, symplectic volume, group representation theory, packing stability, and higher-dimensional algebraic structures, providing analytic, algebraic, and geometric controls on remainder terms and subleading asymptotics.

1. Equivariant Weyl Laws and Symplectic Reductions

Equivariant symplectic Weyl laws arise when invariant elliptic pseudodifferential operators act on compact Riemannian manifolds $M$ equipped with effective, isometric actions by compact Lie groups $G$ (Ramacher, 2010). The relevant spectral counting function $N_x(\lambda)$, which counts eigenvalues (or rather $Q$-eigenvalues for $Q=P^{1/m}$) in an isotypic component, admits an asymptotic expansion of the form: $$N_x(\lambda) = \frac{d_x [T_x|_H : \mathbf{1}]}{(2\pi)^{n-k}} \lambda^{\frac{n-k}{m}} \operatorname{vol}\left[\frac{\Omega \cap S^*M}{G}\right] + O\left(\lambda^{\frac{n-k-1}{m}} (\log \lambda)^{\Lambda}\right)$$ where $n = \dim M$, $k = \dim$ principal $G$-orbit, $d_x$ is the dimension of the irreducible representation indexed by $x$, $[T_x|_H : \mathbf{1}]$ denotes multiplicity of the trivial representation in the restriction of the isotropy representation to $H$, the principal isotropy subgroup, and $m$ is the order of the operator.

The leading term quantifies the reduced symplectic volume of the quotient $(\Omega \cap S^*M)/G$ given by the zero-level set of the momentum map. The remainder term is sensitive to the singularity structure of the group action and uses logarithmic factors depending on the complexity of the resolution of singularities. Resolution techniques involve iterative monoidal transformations—blow-ups centered on isotropy bundles over maximally singular orbits—yielding monomialization of phase functions and enabling stationary phase analysis over clean critical sets.

This methodology connects spectral geometry and symplectic reduction, ensuring that the representation-theoretic and geometric features of $G$-actions directly influence both the leading and remainder terms in the Weyl law. The explicit factorization and analysis illuminate how symplectic volume quantization emerges from invariant operator spectra.

2. Asymptotic Laws for Non-Selfadjoint Operators on Integrable Symplectic Manifolds

When non-selfadjoint perturbations $i\epsilon Q$ are added to selfadjoint semiclassical pseudodifferential operators with completely integrable classical flows in dimension two, the distribution of complex eigenvalues is governed by the geometry of invariant Lagrangian tori—particularly those which satisfy Diophantine conditions (Hitrik et al., 2011). The central asymptotic is: $$N(R; h) = (2\pi h)^{-2} \int_{\Omega(R)} \mu(d\rho) + O(h^{-2} \cdot o(1))$$ where $N(R; h)$ counts eigenvalues within the spectral rectangle $R$, $\mu$ is the Liouville symplectic measure, and the domain $\Omega(R)$ is defined by the joint constraint that the principal symbol $p(\rho)$ lies in a spectral interval and the torus-average $\langle q \rangle(\rho)$ of the perturbation $q$ is contained within prescribed bounds.

These laws rely on action-angle coordinates, allowing for normal form reductions, and exploit the ergodic properties of the Diophantine tori, wherein long-time averages of the perturbation along classical trajectories determine the imaginary parts of eigenvalues. The counting function thus reduces to a measure of phase space regions where both energy and averaged perturbation satisfy given bounds, reflecting the role of symplectic structure and dynamics in spectral asymptotics.

3. Noncommutative Symplectic Weyl Laws and Volume Spectrality

In the noncommutative geometry setting, Weyl laws extend to quantized tori such as $\mathbb{T}_{\theta}^2$ equipped with general translation-invariant conformal structures and Weyl conformal factors (Fathizadeh et al., 2011). The key result is: $$N(X) \sim [S(T) \cdot t(k^{-2})] \cdot X \quad \text{as} \ X \to \infty$$ where $N(X)$ counts the number of Laplacian eigenvalues below level $X$, $S(T)$ reflects the conformal parameter, and $t(k^{-2})$ generalizes the volume integration over the Weyl factor.

Dixmier traces and noncommutative residues coincide for order $-2$ pseudodifferential operators, confirming that noncommutative integrals computed via spectral methods remain consistent with symbol calculus. The heat kernel expansion and spectral asymptotics in this setting "hear" the volume and other geometric invariants of the noncommutative space, reinforcing the deep connection between noncommutative phase spaces, their symplectic measure, and quantum spectral asymptotics.

4. Weyl Harish-Chandra Integration for Symplectic and Super Lie Algebras

Weyl Harish-Chandra integration formulas generalize to reductive dual pairs acting on symplectic spaces $W$ through decompositions into orbital integrals over Cartan subspaces (McKee et al., 2011). The formula,

$$\int_{ST} \varphi(x) dx = \sum_{h_T} \frac{1}{|W(S, h_T)|} \int_{h_T} J(x) \left[\int_{S/S_{h_T}} \varphi(g \cdot (x + y)) d(gS_{h_T})\right] dx$$

relates integration over the odd part $ST$ of a Lie superalgebra to summations over orbits of Cartan subspaces.

Crucially, almost semisimple elements are shown to be dense in $ST$, and refined estimates for orbital integrals along these subspaces yield Schwartz decay properties necessary for Plancherel formulas and Weyl laws. Such analysis establishes symplectic generalizations of classical Weyl law for semisimple Lie groups via techniques suited to super, symplectic, and geometric representation-theoretic settings.

5. Packing Stability and Subleading Asymptotics of Symplectic Weyl Laws

Packing stability asserts the asymptotic ability to fill any compact, connected symplectic $4$-manifold with smooth boundary by symplectic balls or ellipsoids, and this geometric flexibility governs the sharpness of subleading error estimates in symplectic Weyl laws (Edtmair, 18 Sep 2025). For spectral invariants such as Embedded Contact Homology (ECH) capacities, the Weyl law

$$c_k(X) = 2\sqrt{\operatorname{vol}(X) \cdot k} + e_k(X)$$

admits optimal error terms $e_k(X) = O(1)$ for smooth domains, contrasting earlier bounds of order $O(k^{1/4})$ in less regular cases. Similar sharp estimates hold for periodic Floer homology and link spectral invariants, which satisfy

$$c_{L_d}(H) = A^{-1} \int_{[0,1] \times \Sigma} H \, dt \wedge \omega + O(d^{-1})$$

reflecting decaying corrections as the number of link components increases.

The algebraic structure of Hamiltonian diffeomorphism groups, especially Banyaga's perfectness and simplicity properties, underpins both packing results and spectral error bounds. Quantitative fragmentation and gluing techniques enable decomposition into ellipsoids and polydisks, while singular counterexamples with rough boundaries demonstrate failures of packing stability accompanied by divergent error terms in the spectral asymptotics.

6. Algebraic Structures: Symplectic Differential Reduction Algebras and Generalized Weyl Algebras

Symplectic Weyl laws also manifest in algebraic settings involving the representation theory of symplectic Lie algebras and their differential reduction algebras (Hartwig et al., 24 Mar 2024). Here, canonical realizations of $\mathfrak{sp}(2n)$ within the $n$th Weyl algebra yield reduction algebras presented as generalized Weyl algebras (GWAs), specifically of skew-affine type.

In these constructions, the reduction algebra $D(\mathfrak{sp}(4))$ is interpreted as a GWA $B(\sigma, t)$, where $B$ is a polynomial ring over a commutative Cartan subring, and automorphisms $\sigma_i$ act linearly-affine on the variables. Semiclassical limits effectively recover standard Weyl relations, indicating that spectral multiplicities or asymptotic counting in representation-theoretic contexts retain underlying symplectic features. Weight module classification and explicit commutation relation deformation underlie computational approaches to multiplicity estimates and spectral asymptotics in the symplectic context.

7. Symplectic Weyl Laws in Quantum Maps, Integrable Systems, and Packing Geometry

The subject incorporates quantum dynamical systems, such as open quantum baker’s maps, where the asymptotic count of eigenvalues in the semiclassical limit is governed by the fractal dimension $\delta$ of trapped sets: $\mathcal{N}_N(\nu) = O(N^{\delta})$ for $h = (2\pi N)^{-1}$, with further refinements for Gevrey cutoffs yielding explicit dependence on subleading parameters (Li, 2022). In mean-field interacting fermionic systems, semiclassical Weyl laws confirm convergence to the Thomas-Fermi ground state: $$\rho_{\text{TF}}(x) = \left(\frac{B_{\mathbb{R}^d}(0,1)}{(2\pi)^d}\right) [E - V(x) - (w * \rho_{\text{TF}})(x)]_+^{d/2}$$ showing that leading spectral asymptotics are always governed by symplectic phase-space volumes, even in interacting or deformed quantization settings (Nguyen, 2023). Packing stability for symplectic manifolds with smooth boundaries enforces bounded subleading error terms in spectral invariants, while geometric failures in rough boundary cases yield divergent error terms.

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Collectively, Symplectic Weyl Laws synthesize classical spectral theory and modern geometric, algebraic, and quantum principles. The leading asymptotics universally encode symplectic volumes—often of reduced, quotient, or noncommutative spaces—while subleading asymptotics and error terms reflect the fine structure of group actions, boundary conditions, packing flexibility, and algebraic deformations. These laws furnish analytic and structural foundations for regularity statements, quantization procedures, and representation-theoretic multiplicities across symplectic geometry, mathematical physics, and invariant theory.