Symplectic Variants of Weyl's Law

• Symplectic variants of Weyl's law are asymptotic formulas for spectral quantities that refine classical eigenvalue counting by incorporating symplectic geometry and group symmetries.
• They employ advanced techniques such as symplectic reduction, monoidal transformations, and topological recursion to capture refined phase-space and quantum correction data.
• Applications span random matrix theory, noncommutative spaces, and Hamiltonian dynamics, yielding improved remainder estimates and insights into spectral geometry.

Symplectic variants of Weyl’s law refer to asymptotic formulas for spectral quantities—such as eigenvalue counting functions or symplectic capacity invariants—that are adapted to the geometric, algebraic, and dynamical structures inherent in symplectic manifolds and systems with strong symmetry, noncommutative, quantum, or integrable features. These generalizations arise in settings where the classical phase-space volume interpretation via symplectic forms, group actions, or quantum corrections becomes central, and where the leading and subleading terms encode refined symplectic or equivariant data. Symplectic variants are now recognized as essential tools in spectral geometry, Hamiltonian dynamics, random matrix theory, quantum field theory, and noncommutative analysis.

1. Equivariant Weyl Laws and Symplectic Reduction

When a compact Lie group $G$ acts effectively and isometrically on a manifold $M$, and one considers an invariant elliptic pseudodifferential operator $P_0$, the classical Weyl law is refined in several ways. First, the spectrum of $P$ splits into isotypic components indexed by irreducible representations (characters) of $G$, so the $L^2$ space becomes $L^2(M) \cong \bigoplus_{x \in \hat{G}} L^2(M)(x)$. For each such $x$, the reduced spectral counting function

$$N_x(\lambda) = \frac{1}{d_x}\sum_{t\leq\lambda} \mathrm{mult}_x(t)$$

admits an asymptotic expansion

$$N_x(\lambda) = [T_x|_H:1]^{-1} (2\pi)^{-(n-k)} \mathrm{Vol}((\mathrm{reg}\,S^*M)/G)\, \lambda^{(n-k)/m} + O\left( \lambda^{(n-k-1)/m}(\log\lambda)^A \right)$$

where $k$ is the dimension of a principal $G$-orbit, $H$ is a principal isotropy group, $T_x|_H:1$ is the multiplicity of the trivial representation in the restriction to $H$, and $A$ depends on the singular strata (Ramacher, 2010, Ramacher, 2015). The leading term involves the symplectic volume of the quotient space $(S^*M\cap\Omega)/G$, with $\Omega = J^{-1}(0)$ the zero level of the momentum map $J:T^*M\rightarrow\mathfrak{g}^*$; the reduction expresses the refined phase-space structure, central to the symplectic variant.

The proof relies on resolving singularities of the critical set associated to the non-free action—via sequences of blow-ups (monoidal transformations) along orbit-type strata—so that the phase function in trace formulas admits stationary phase expansion on a “partially monomialized” (clean) set. The appearance of logarithmic factors in the error term quantitatively archives orbit stratification complexity.

2. Symplectic Invariants in Random Matrix Theory and Integrable Systems

A distinct but related symplectic refinement emerges from random matrix theory and its connection with integrable systems. In the Tracy-Widom regime for the maximal eigenvalue distribution in GUE, the full asymptotic expansion of the probability law

$$F_{\rm GUE}(s) = C \exp\left\{ -\frac{|s|^3}{12} - \frac{1}{8}\log|s| + \sum_{g\ge2} \left(-\frac{s}{2}\right)^{3(1-g)} F^g(\Sigma_{\rm TW}) \right\}$$

is expressed in terms of symplectic invariants $F^g(\Sigma_{\rm TW})$ computed via topological recursion on the local spectral curve $\Sigma_{\rm TW}: y^2 = x + (1/x) - 2$ (Borot et al., 2010, Borot et al., 2010). These invariants depend only on the two-form $dx\wedge dy$ and are unchanged under symplectic (canonical) changes of coordinates.

Integrable system techniques show that the Tracy-Widom law also arises from the tau-function of a Painlevé II Lax pair, with the Hamiltonian structure and symplectic form underlying the semiclassical expansion. The leading cubic term parallels the classical Weyl law (volume term); the infinite series of symplectic invariants encode the “quantum corrections,” yielding a symplectic variant of Weyl’s law for the full probability distribution and spectral density.

3. Remainder Estimates and Hamiltonian Recurrence

Remainder estimates in symplectic Weyl laws are closely linked to the recurrence properties of the associated Hamiltonian flow. For an elliptic, semiclassical pseudodifferential operator $P_h$ with symbol $p_0$, the remainder $R_h$ in the counting function

$$N_h[a,b] = (2\pi h)^{-n} \left[ \mathrm{vol}\{ p_0^{-1}[a, b] \} + h(\cdots) + h R_h \right]$$

is estimated by the Liouville volumes of recurrence sets for the symplectic flow $e^{tH_{p_0}}$ (see (Savale, 2023)):

$$|R_h| \le (\nu(\Sigma_a)+\nu(\Sigma_b))T^{-1} + O(\nu(S_{T,\epsilon}^{a,e}) + T^{-2} + h^{1-2\delta})$$

where $S_{T,\epsilon}^{a,e}$ captures points on the energy surface $\Sigma_a$ returning close to their initial state under the flow. For Anosov flows, exponential bounds in terms of maximal expansion rate and entropy are achieved; for specific manifolds—compact Lie groups, surfaces of revolution—explicit polynomial remainder exponents arise, quantifying improvements over classical estimates.

These results indicate a central theme: the dynamical symplectic invariants—recurrence rates, entropy, vanishing orders of return maps—directly refine the remainder in spectral asymptotics.

4. Symplectic Weyl Laws for Noncommutative and Quantum Manifolds

The symplectic framework extends to noncommutative geometry and quantum homogeneous spaces. For the noncommutative two-torus $\mathbb{T}^2_\theta$ with translation-invariant conformal structure and Weyl factor, spectral triple methods yield a Weyl law (Fathizadeh et al., 2011):

$N(\lambda)\sim \pi\, S(T)\, \tau(k^{-2})\, \lambda$

where $S(T)$ is a geometric factor and $\tau(k^{-2})$ is the noncommutative integral replacing classical volume. Connes’ trace theorem generalizes, with the Dixmier trace equated to the noncommutative residue for pseudodifferential operators of order $-2$. The symplectic measure is replaced by traces over noncommutative algebras, preserving the volume-theoretic spirit.

Similarly, for quantized flag manifolds and compact quantum groups, Weyl-type zeta functions constructed from central Casimir elements recover both dimension and volume in the quantum setting (Matassa, 2014). Proportionality with the Haar state and first singularity at the classical dimension are demonstrated, showing that spectral data encode geometric and symplectic invariants of the underlying quantum spaces.

5. Packing Stability, Subleading Asymptotics, and Hamiltonian Group Structure

Recent advances link sharp symplectic Weyl law error terms to global packing properties and to the algebraic flexibility of Hamiltonian diffeomorphism groups. In compact symplectic $4$-manifolds with smooth boundary, “packing stability” holds: the volume can be filled by a single ellipsoid, and corresponding symplectic spectral invariants (ECH capacities, PFH invariants) satisfy

$c_k(X) = 2\sqrt{\mathrm{vol}(X)\, k} + O(1)$

(Edtmair, 18 Sep 2025). The $O(1)$ subleading term is sharp, and packing stability fails for domains with only $C^{1,\alpha}$ boundary—error terms diverge. The proofs exploit deep results on the simplicity and perfectness of Hamiltonian diffeomorphism groups (Banyaga), allowing the decomposition of symplectic cobordisms into elementary ribbon complexes, frusta, and cuboids for effective local packing, which in turn yields spectral control.

This interplay—between symplectic embedding, group-theoretic flexibility, and rigorous asymptotics—demonstrates that boundary regularity, symplectic packing, and the algebraic structure of Hamiltonian groups are decisive for quantitative law refinements.

6. Singular, Noncompact, and Volume-Growth Variants

On singular Riemannian manifolds or noncompact (asymptotically Euclidean) spaces, Weyl’s law must be modified to reflect geometric anomalies. For singular structures with unbounded curvature or infinite volume, the eigenvalue counting function admits

$$N(\lambda) \sim \frac{\omega_n}{(2\pi)^n} \lambda^{n/2} \upsilon(\lambda)$$

where $\upsilon(\lambda)$ encodes anomalous local volume growth near singular sets—this includes factors like $\log\lambda$ or $\exp((\log\lambda)^\alpha)$ (Chitour et al., 2019). For SG-operators with symbol $(1+|x|^2)(1-\Delta)$ on asymptotically Euclidean manifolds, sharp two- and three-term Weyl formulas are obtained with explicit logarithmic corrections, tightly linked to Hamiltonian flow analysis at infinity (Coriasco et al., 2019).

Symplectic geometry underlies these results: the integration measures and flows are symplectic in nature, and the corrections (logarithmic, polynomial) stem from global flow behaviour, recurrence rates, and singularity structure.

7. Non-linear and Volume Spectrum Weyl Laws

The scope of symplectic Weyl laws now includes nonlinear min-max volume spectra, as established for 1-cycles in manifolds, confirming Gromov’s conjecture (Staffa, 30 Oct 2024). The Almgren–Pitts widths for sweepouts by 1-cycles satisfy

$$\lim_{p\to\infty} \omega_p^1(M,g)\, p^{-(n-1)/n} = \alpha(n)\, \mathrm{Vol}(M,g)^{1/n}$$

with precise normalization via parametric coarea and isoperimetric inequalities. The analytic machinery uses localization on cubical complex cells and approximation by $\delta$-localized families. Such techniques point to possible symplectic analogues for min-max spectra involving Lagrangian or pseudoholomorphic sweepouts, hinting that similar parametric inequalities and local-to-global inductive methods may underpin the symplectic non-linear spectrum.

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Collectively, the modern theory of symplectic variants of Weyl’s law encompasses spectral asymptotics for operators, noncommutative and quantum spaces, nonlinear volume spectra, random matrix models, and extends to refined remainder estimates that quantify symplectic dynamics, group actions, and packing stability. The field is characterized by the interplay of symplectic reduction, recurrent flow analysis, topological recursion, and quantitative group theory, providing a rich structure for addressing spectral questions in geometric analysis, mathematical physics, and beyond.