Semiclassical measures for quantized elliptic and parabolic symplectic linear maps of the multidimensional torus
Abstract
We study the semiclassical measures associated to the quantum map on the torus obtained by quantizing integer symplectic linear maps, focusing on periodic matrices and certain parabolic matrices. More precisely, we consider the time-one map of the completely integrable flows generated by Hamiltonians of the form H(x, ξ) = Q(ξ)/2 where Q is an integral quadratic form. We prove that the set of semiclassical measures associated to a periodic matrix A coincides with the set of invariant probability measures of A. The semiclassical measures of parabolic matrices on T² are explicitly characterized by proving that they have at most three nonzero Fourier coefficients in position. A partial quantum ergodicity statement for parabolic matrices is obtained by proving a sharp bound on the quantum variance and some examples are built in higher dimensions. In particular, starting from T⁴ there always exist semiclassical measures whose projection on the position has only nonzero Fourier coefficients. Moreover, if Q is isotropic over the rationals, then there exist semiclassical measures whose projection on the position concentrate on some subtori. A link is made with the semiclassical measures of the Laplacian on the torus.
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BibTeX
@misc{ferraridominguez2026semiclassical,
title = {{Semiclassical measures for quantized elliptic and parabolic symplectic linear maps of the multidimensional torus}},
author = {Vincent Ferrari-Dominguez},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26382},
eprint = {2610.26382},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05805563v1},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Semiclassical measures for quantized elliptic and parabolic symplectic linear maps of the multidimensional torus
AU - Vincent Ferrari-Dominguez
PY - 2026
DA - 2026/10/19
PB - HAL
DO - 10.48550/arxiv.2610.26382
UR - https://hal.science/hal-05805563v1
AB - We study the semiclassical measures associated to the quantum map on the torus obtained by quantizing integer symplectic linear maps, focusing on periodic matrices and certain parabolic matrices. More precisely, we consider the time-one map of the completely integrable flows generated by Hamiltonians of the form H(x, ξ) = Q(ξ)/2 where Q is an integral quadratic form. We prove that the set of semiclassical measures associated to a periodic matrix A coincides with the set of invariant probability measures of A. The semiclassical measures of parabolic matrices on T² are explicitly characterized by proving that they have at most three nonzero Fourier coefficients in position. A partial quantum ergodicity statement for parabolic matrices is obtained by proving a sharp bound on the quantum variance and some examples are built in higher dimensions. In particular, starting from T⁴ there always exist semiclassical measures whose projection on the position has only nonzero Fourier coefficients. Moreover, if Q is isotropic over the rationals, then there exist semiclassical measures whose projection on the position concentrate on some subtori. A link is made with the semiclassical measures of the Laplacian on the torus.
ER -