# Upper bound on the ground state energy of the two-component charged Bose gas with arbitrary masses

### Andreas Schön

Ludwig-Maximilians-Universität München, Munich, Germany### Jan Philip Solovej

University of Copenhagen, Denmark

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## Abstract

The upper bound on the ground state energy of the two-component charged Bose gas derived in [Comm. Math. Phys. 266 (2006), 797–818] is extended to the more general case, where the mass of the positive bosons can be different from the mass of the negative bosons. The analysis in principle is quite similar, however the formerly scalar problem becomes two-dimensional (one dimension for the $+$ and one for the $−$ case). This leads to a non-linear $2×2$-matrix minimization problem. Its minimizer is calculated and results in a non-linear mass-dependent coefficient in the upper bound. These new results agree in the equal-mass case with the formerly known results. It is also discussed how the result here interpolates between the equal mass case and the case where one mass is infinite.