On the Lebesgue Component of Semiclassical Measures for Abelian Quantum Actions
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916751687548928 |
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| author | Rivière, Gabriel Wolf, Lasse L. |
| author_facet | Rivière, Gabriel Wolf, Lasse L. |
| contents | For a large class of symplectic integer matrices, the action on the torus extends to a symplectic $\mathbb{Z}^r$-action with $r\geq 2$. We apply this to the study of semiclassical measures for joint eigenfunctions of the quantization of the symplectic matrices of the $\mathbb{Z}^r$-action. In the irreducible setting, we prove that the resulting probability measures are convex combinations of the Lebesgue measure with weight $\geq 1/2$ and a zero entropy measure. We also provide a general theorem in the reducible case showing that the Lebesgue components along isotropic and symplectic invariant subtori must have total weight $\geq 1/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16472 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Lebesgue Component of Semiclassical Measures for Abelian Quantum Actions Rivière, Gabriel Wolf, Lasse L. Mathematical Physics Analysis of PDEs Spectral Theory For a large class of symplectic integer matrices, the action on the torus extends to a symplectic $\mathbb{Z}^r$-action with $r\geq 2$. We apply this to the study of semiclassical measures for joint eigenfunctions of the quantization of the symplectic matrices of the $\mathbb{Z}^r$-action. In the irreducible setting, we prove that the resulting probability measures are convex combinations of the Lebesgue measure with weight $\geq 1/2$ and a zero entropy measure. We also provide a general theorem in the reducible case showing that the Lebesgue components along isotropic and symplectic invariant subtori must have total weight $\geq 1/2$. |
| title | On the Lebesgue Component of Semiclassical Measures for Abelian Quantum Actions |
| topic | Mathematical Physics Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2505.16472 |