Weyl asymptotics for pseudodifferential operators in a discrete setting
Markus Klein, Enrico Reiss, Elke Rosenberger
TL;DR
This work establishes sharp Weyl asymptotics for a broad class of self-adjoint discrete pseudodifferential operators acting on the scaled lattice ℓ^2(εℤ^d). By identifying the relevant phase space as ℝ^d × 𝕋^d and constructing a semiclassical time-evolution parametrix that respects ξ-periodicity, the authors derive a leading Weyl term N([α,β];ε) ≈ (vol_T(a_0^{-1}([α,β])))/(2π ε)^d with an O(ε) remainder, and improve the remainder to a positive power when α and β are non-critical values of a_0. The analysis blends invertibility and functional calculus on the discrete setting, trace-class criteria, and a semiclassical Fourier integral operator framework to connect spectral counts with phase-space volumes via Liouville measure on level sets of the principal symbol. These results extend semiclassical Weyl theory to a broad class of difference operators on scaled lattices and provide technical tools applicable to metastability and spectral theory of Markov generators in high-dimensional discrete spaces. The methods, including a periodic-discrete Helffer–Sjöstrand calculus and a careful time-parametrix construction, offer a rigorous pathway to refined spectral asymptotics in discrete-geometry contexts with finite accessible phase space.
Abstract
We prove a sharp Weyl estimate for the number of eigenvalues belonging to a fixed interval of energy of a self-adjoint difference operator acting on $\ell^2(ε\mathbb{Z}^d)$ if the associated symplectic volume of phase space in ${\mathbb R}^d \times {\mathbb T}^d$ accessible for the Hamiltonian flow of the principal symbol is finite. Here $ε$ is a semiclassical parameter. Our proof depends crucially on the construction of a good semiclassical approximation for the time evolution induced by the self-adjoint operator on $\ell^2(ε\mathbb{Z}^d)$. This extends previous semiclassical results to a broad class of difference operators on a scaled lattice.