Monotonicity of Quantum Ground State Energies: Bosonic Atoms and Stars

The N-dependence of the non-relativistic bosonic ground state energy ? ^B (N) is studied for quantum N-body systems with either Coulomb or Newton interactions. The Coulomb systems are “bosonic atoms,” with their nucleus fixed, and it is shown that $\mathcal {E}_{{C}}^{{B}}(N)/\mathcal {P}_{{C}}(N)$ grows monotonically in N>1, where ? _C (N)=N ²(N?1). The Newton systems are “bosonic stars,” and it is shown that when the Bosons are centrally attracted to a fixed gravitational “grain” of mass M>0, and N>2, then $\mathcal {E}_{{N}}^{{B}}(N;M)/\mathcal {P}_{\!{N}}(N)$ grows monotonically in N, where ? _N (N)=N(N?1)(N?2); in the translation-invariant problem (M=0), it is shown that when N>1 then $\mathcal {E}_{{N}}^{{B}}(N;0)/\mathcal {P}_{{C}}(N)$ grows monotonically in N, with ? _C (N) from the Coulomb problem. Some applications of the new monotonicity results are discussed.