Weyl asymptotics for pseudodifferential operators in a discrete setting

2025/10/13 by Klein, Markus, Reiss, Enrico, Rosenberger, Elke
#35S05 (Primary) 39A70 #41A60 #47A10 #81Q10 (Secondary) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2510.11193

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 ℓ²(εℤ^d) if the associated symplectic volume of phase space in \mathbb R^d × \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 ℓ²(εℤ^d). This extends previous semiclassical results to a broad class of difference operators on a scaled lattice.

Citations

• Continuum limit of resonances for discrete Schrödinger operators
• Modified wave operators for discrete Schrödinger operators with long-range perturbations
• Agmon-type estimates for a class of jump processes
• Metastability and low lying spectra in reversible Markov chains
• Metastability in stochastic dynamics of disordered mean-field models
• Analyticity properties in scattering and spectral theory for Schrödinger operators with long-range radial potentials
• The spectral function of an elliptic operator
• Fourier integral operators. II

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• Reduced Weyl asymptotics for pseudodifferential operators on bounded domains II. The compact group case
• Singular equivariant asymptotics and Weyl's law