On the Lebesgue Component of Semiclassical Measures for Abelian Quantum Actions

2025/05/22 by Rivière, Gabriel, Wolf, Lasse L. · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2505.16472

Abstract

For a large class of symplectic integer matrices, the action on the torus extends to a symplectic ℤ^r-action with r≥ 2. We apply this to the study of semiclassical measures for joint eigenfunctions of the quantization of the symplectic matrices of the ℤ^r-action. In the irreducible setting, we prove that the resulting probability measures are convex combinations of the Lebesgue measure with weight ≥ 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 ≥ 1/2.

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