Weyl Type Asymptotics in Spectral Analysis

• Weyl type asymptotic formulas are a set of results that quantify the leading behavior of eigenvalue counts and spectral densities in relation to geometric measures.
• They extend classical asymptotic laws to operators with boundaries, noncommutative structures, and higher-order corrections such as logarithmic terms.
• Analytical tools like microlocal analysis, zeta function methods, and Tauberian theorems are pivotal in extracting precise asymptotic expansions.

Weyl type asymptotic formulas are foundational results in spectral theory that describe the leading-order asymptotic behavior of quantities such as eigenvalue counting functions, spectral measures, or resonance counts associated with differential, difference, or pseudodifferential operators in mathematical physics and geometry. These formulas have profound connections to microlocal analysis, random matrix theory, number theory, and noncommutative geometry. Despite their diverse manifestations, Weyl type asymptotic formulas universally relate the "large-scale" spectral properties of an operator to geometric or combinatorial invariants of the underlying system.

1. Classical Weyl Law and Its Extensions

The prototype Weyl law describes the asymptotic distribution of eigenvalues $\lambda_j$ of the Dirichlet Laplacian $-\Delta$ on a bounded domain $X \subset \mathbb{R}^d$. The counting function $N(\lambda) = \#\{\lambda_j < \lambda\}$ satisfies

$$N(\lambda) = (2\pi)^{-d}\omega_d \operatorname{vol}(X) \lambda^{d/2}(1+o(1)), \quad \lambda\to\infty,$$

where $\omega_d$ is the volume of the unit ball in $\mathbb{R}^d$ (Ivrii, 2016). This leading term, the "Weyl principal part," directly connects the asymptotic spectral density to the phase space volume. Extensions to operators with boundary, varying coefficients, or magnetic/Dirac operators introduce correction terms, such as $\lambda^{(d-1)/2}$ effects from the boundary or periodic orbit structure.

In automorphic and arithmetic settings, Selberg-type and Weyl's laws for the Laplacian or Hecke operators count automorphic cusp forms, with leading asymptotics determined by the spectral parameter and geometric invariants of the locally symmetric space. Generalizations to non-spherical representations incorporate explicit multiplicities, e.g., a $\dim(\tau)$ factor for cusp forms of arbitrary $K_\infty$ type (Maiti, 2022).

2. Formulas for Spectral Density and Orthonormal Polynomial Asymptotics

Weyl–Titchmarsh type formulas relate the asymptotic behavior of orthonormal polynomials or generalized eigenfunctions to spectral densities of discrete and continuous Schrödinger operators. For a discrete Schrödinger operator $J$ with a Wigner–von Neumann potential, the orthogonal polynomials $P_n(\lambda)$ exhibit asymptotics involving a Jost function $F(z)$. The spectral density $p'(\lambda)$ at non-critical $\lambda$ is given by

$$p'(\lambda) = \frac{\sqrt{4-\lambda^2}}{2\pi |F(z)|^2}, \quad \lambda = z + 1/z$$

(Janas et al., 2010). This approach generalizes to discrete Hermite-type operators, yielding, under perturbations, the density

$$\rho'(\lambda) = \frac{e^{-\lambda^2/2}}{\sqrt{2\pi}|F(\lambda)|^2}$$

(Simonov, 2010). The central object in these formulas is the Jost function, which encapsulates the effect of the potential (or perturbation) as a coefficient in the leading asymptotics of the orthonormal polynomials.

These techniques extend to situations with subtle spectral effects, such as the presence of critical (resonant) points, or slowly decaying/oscillatory potentials (e.g., Wigner–von Neumann), and provide a robust framework to interpret the absolutely continuous spectrum, embedded eigenvalues, and modifications of the spectral density.

3. Weyl Asymptotics in Pseudodifferential and Bisingular Operators

For general (pseudo)differential operators, the eigenvalue counting function admits more sophisticated expansions. On products of manifolds, positive self-adjoint bisingular operators have counting functions of the form

$$N_A(\lambda) \sim C'_A\,\lambda^\delta \log\lambda + C_A\,\lambda^\delta + o(\lambda^\delta)$$

where the presence of a logarithmic term is tied to a double pole in the meromorphic continuation of the associated zeta function $\zeta(A,z)$ (Battisti, 2010). An influential example is the spectral connection to the Dirichlet divisor problem: for $A = -\Delta_1 \otimes -\Delta_2$ on $S^1 \times S^1$, $N_A(\lambda)$ is asymptotically proportional to the classical divisor sum $D(\sqrt{\lambda}) \sim 2x\log x + (2\gamma-1)x$, directly relating spectral asymptotics to number-theoretic structure.

Weighted local Weyl laws extend these results to spectral projectors and more general kernels: $$K_L^s(x, y) = \sum_{0<\lambda_k\leq L}\lambda_k^{-s} e_k(x)\overline{e_k(y)}$$ with explicit diagonal scaling for subcritical $s < n/m$, and a logarithmic singularity in the critical case $s = n/m$ (Rivera, 2018).

4. Noncommutative, Foliation, and Adiabatic Limit Generalizations

Noncommutative geometry provides a setting for "noncommutative Weyl formulas," for example in the spectral asymptotics of Laplacians under adiabatic rescaling of foliated Riemannian manifolds: $$\operatorname{tr}f(\Delta_\varepsilon) = \frac{1}{(2\pi)^q} \operatorname{tr}_{\mathcal{F}_N} f(\sigma(\Delta_\varepsilon)) + o(\varepsilon^{-q}),\quad \varepsilon\to 0$$ (Kordyukov, 2010). Here, $\operatorname{tr}_{\mathcal{F}_N}$ is a Connes-inspired noncommutative trace over the leaf space's cotangent bundle, reflecting the lack of a classical phase space measure for singular foliations.

In the context of spectral triples in noncommutative geometry,

$$\lim_{h\downarrow 0} h^p\operatorname{Tr}(\chi_{(-\infty,0)}(h^2 D^2+V)) = \int V_-^{p/2}|ds|^p$$

establishes that semiclassical eigenvalue asymptotics encode a noncommutative integral (McDonald et al., 2021).

5. Weyl-Type Counting: Resonances, Walks, and Higher-Order Problems

Weyl type asymptotics are not confined to self-adjoint spectra, but also appear in resonance counting, e.g., Schrödinger operators with point interactions. The resonance counting function follows

$N_{\mathcal{H}}(R) = (W/\pi)R + O(1),$

where $W$ is a geometric invariant of the configuration of delta-impurities, generically equal to the maximal sum of inter-point distances over all permutations (Albeverio et al., 2018).

For combinatorial models, such as zero-drift lattice walks confined to a Weyl chamber of type $A$ (and the associated "vicious walker" models), the number of $n$-step walks between fixed points $u$ and $v$ admits precise asymptotics: $$P_n^+(u \to v) \sim S(1, ..., 1)^n n^{k^2/2} \prod_{1\leq j<m\leq k} (u_m-u_j)(v_m-v_j),$$ where $S$ is the symmetry-invariant step generating function, and the pre-factor includes Vandermonde determinants reflecting particle repulsion (Feierl, 2018).

Higher-order boundary value problems, such as biharmonic Steklov eigenvalues under Neumann-type boundary conditions, exhibit Weyl-type formulas for their counting functions: $$A(T) \sim W_{n-1} (3/16)^{-(n-1)} T^{n-1} \int_{\partial D} ds,$$ with $W_{n-1}$ the volume of the unit $(n-1)$-ball and $ds$ the boundary measure (Liu, 2010, Liu, 2011). These expansions are determined by the order-1 principal symbol of associated pseudodifferential boundary operators, a crucial difference from classical Laplacian asymptotics.

6. Analytical Tools: Zeta Functions, Commutator Criteria, and Microlocal Methods

Weyl type formulas are derived using diverse analytical machinery. The meromorphic continuation of zeta functions, Tauberian theorems (including for noncommuting operators (McDonald et al., 2021)), and the spectral analysis of pseudodifferential symbols underpin the precise extraction of leading and subleading terms. The Beals characterization theorem connects the boundedness of iterated commutators to the existence of global Weyl symbols, with explicit integral formulas involving Gaussian measures yielding asymptotic expansions suitable even for infinite-dimensional settings (Amour et al., 2018).

Semiclassical and microlocal analysis, particularly in the paper of propagation of singularities and short-time wave kernel parametrices, provides the underlying estimates and reductions for spectral projector kernels, boundary corrections, and the impact of periodic trajectories. For instance, the analysis of the flow on the "corner" in the double compactification of the cotangent bundle is essential in extracting log-type correction terms on asymptotically Euclidean manifolds (Coriasco et al., 2019).

7. Broader Implications and Universality

Weyl type asymptotic formulas establish "universal" connections between spectral quantities and underlying geometric, combinatorial, or even noncommutative invariants. Their universality is seen across classical analysis, combinatorics, number theory (e.g., Dirichlet divisor problem), operator algebras, and quantum field theory. Modern research pushes these results beyond classical smooth manifolds and self-adjoint spectra, into domains with singularities, boundaries, or even noncommutative topology, with new analytic tools (noncommutative integration, singular value Tauberian theory, etc.) required for their full elucidation.

In all these contexts, Weyl type asymptotic formulas remain central, providing the leading asymptotic "fingerprint" of the underlying space, operator, or model and encoding deep invariants that govern the density, distribution, and fluctuations of spectral data.