We study asymptotically and numerically the fundamental gaps (i.e. the difference between the first excited state and the ground state) in energy and chemical potential of the Gross-Pitaevskii equation (GPE) -- nonlinear Schrodinger equation with cubic nonlinearity -- with repulsive interaction under different trapping potentials including box potential and harmonic potential. Based on our asymptotic and numerical results, we formulate a gap conjecture on the fundamental gaps in energy and chemical potential of the GPE on bounded domains with the homogeneous Dirichlet boundary condition, and in the whole space with a convex trapping potential growing at least quadratically in the far field. We then extend these results to the GPE on bounded domains with either the homogeneous Neumann boundary condition or periodic boundary condition.
We prove the convergence of the thermodynamic functions of a free boson gas for ad-dimensional (d=3,4,...) van Hove sequence of convex regions. The thermodynamic functions behave singularly at a critical densityρ c which is independent of the geometrical details of the sequence. We are led to define a second critical densityρ m depending on the geometrical details of the sequence. For densities betweenρ c andρ m none of the single particle states is macroscopically occupied. We derive a sufficient condition on the sequence such thatρ m =ρ c .
We report enhancement of the mechanical stability of graphene through a one-step method to disperse gold nanoparticles on the pristine graphene without any added agent.