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The nonlinear Schr\"odinger equation for orthonormal functions: I. Existence of ground states

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arxiv 2002.04963 v2 pith:LDOTOGSP submitted 2020-02-12 math.AP math-phmath.MPmath.SP

The nonlinear Schr\"odinger equation for orthonormal functions: I. Existence of ground states

classification math.AP math-phmath.MPmath.SP
keywords equationexistencefunctionsgroundlargenonlinearodingerorthonormal
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We study the nonlinear Schr\"odinger equation for systems of $N$ orthonormal functions. We prove the existence of ground states for all $N$ when the exponent $p$ of the non linearity is not too large, and for an infinite sequence $N_j$ tending to infinity in the whole range of possible $p$'s, in dimensions $d\geq1$. This allows us to prove that translational symmetry is broken for a quantum crystal in the Kohn-Sham model with a large Dirac exchange constant.

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