Quantum unique ergodicity for magnetic Laplacians on T^2

Given a smooth integral two-form and a smooth potential on the flat torus of dimension 2, we study the high energy properties of the corresponding magnetic Schrödinger operator. Under a geometric condition on the magnetic field, we show that every sequence of high energy eigenfunctions satisfies the quantum unique ergodicity property even if the Liouville measure is not ergodic for the underlying classical flow (the Euclidean geodesic flow on the 2-torus).

💡 Research Summary

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The paper investigates high‑energy eigenfunctions of magnetic Schrödinger operators on the flat two‑dimensional torus (\mathbb T^{2}=\mathbb R^{2}/\mathbb Z^{2}). The authors consider a smooth, real‑valued 2‑form (B) (the magnetic field) satisfying the integral quantisation condition (\int_{\mathbb T^{2}} B\in 2\pi\mathbb Z) and a smooth real potential (V). The magnetic Laplacian is defined as
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