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Non-gaussian waves in Seba's billiard

The Seba billiard, a rectangular torus with a point scatterer, is a popular model to study the transition between integrability and chaos in quantum systems. Whereas such billiards are classically essentially integrable, they may display features such as quantum ergodicity [KU] which are usually associated with quantum systems whose classical dynamics is chaotic. Seba proposed that the eigenfunctions of toral point scatterers should also satisfy Berry's random wave conjecture, which implies that the semiclassical moments of the eigenfunctions ought to be Gaussian. We prove a conjecture of Keating, Marklof and Winn who suggested that Seba billiards with irrational aspect ratio violate the random wave conjecture. More precisely, in the case of diophantine tori, we construct a subsequence of eigenfunctions of essentially full density and show that its semiclassical moments cannot be Gaussian.