Weyl asymptotics for pseudodifferential operators in a discrete setting
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908589314015232 |
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| author | Klein, Markus Reiss, Enrico Rosenberger, Elke |
| author_facet | Klein, Markus Reiss, Enrico Rosenberger, Elke |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_11193 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weyl asymptotics for pseudodifferential operators in a discrete setting Klein, Markus Reiss, Enrico Rosenberger, Elke Spectral Theory Mathematical Physics 47A10, 35S05 (Primary) 39A70, 41A60, 81Q10 (Secondary) 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. |
| title | Weyl asymptotics for pseudodifferential operators in a discrete setting |
| topic | Spectral Theory Mathematical Physics 47A10, 35S05 (Primary) 39A70, 41A60, 81Q10 (Secondary) |
| url | https://arxiv.org/abs/2510.11193 |