  We establish an a priori error analysis for the lowest-order Raviart-Thomas
finite element discretisation of the nonlinear Gross-Pitaevskii eigenvalue
problem. Optimal convergence rates are obtained for the primal and dual
variables as well as for the eigenvalue and energy approximations. In contrast
to conformal approaches, which naturally imply upper energy bounds, the
proposed mixed discretisation provides a guaranteed and asymptotically exact
lower bound for the ground state energy. The theoretical results are
illustrated by a series of numerical experiments.
