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Semi-classical measures for inhomogeneous Schrödinger equations on tori

The purpose of this note is to investigate the high frequency behaviour of solutions to linear Schrödinger equations. More precisely, Bourgain and Anantharaman-Macia proved that any weak-* limit of the square density of solutions to the time dependent homogeneous Schrödinger equation is absolutely continuous with respect to the Lebesgue measure on $R\times T^d$. Our contribution is that the same result automatically holds for non homogeneous Schrödinger equations, which allows for abstract potential type perturbations of the Laplace operator.