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arXiv · 2610.26382

Semiclassical measures for quantized elliptic and parabolic symplectic linear maps of the multidimensional torus

Vincent Ferrari-Dominguez
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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 T${}^2$ 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 T___ 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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Vincent Ferrari-Dominguez. 2026-10-19. Semiclassical measures for quantized elliptic and parabolic symplectic linear maps of the multidimensional torus. https://arxiv.org/abs/2610.26382

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