Monotonicity of quantum ground state energies: Bosonic atoms and stars
For quantum \(N\)-body systems the non-relativistic bosonic ground state energy \(E^B(N)\) is studied with either Coulomb or Newton interaction. The Coulomb system are bosonic atoms with fixed nucleus, the quantity \(E^B_C(N)/P_C(N)\) grows monotonically with \(N>1\), where \(P_C(N)= N^2(N- 1)\). The Newton systems are bosonic stars. It is shown that when the bosons are centrally attracted to a fixed gravitational grain of mass \(M>0\) and \(N>2\), then \(E_N^B(N,M)/P_N(N)\) grows monotonically with \(N\), where \(P_N(N)= N(N- 1)(N- 2)\). In the translation invariant problem -- \((M= 0)\) -- it is shown that for \(N>1\) the quantity \(E^B_N(N; 0)/P_C(N)\) grows monotonically with \(N\), here \(P_C(N)\) is the expression for the Coulomb problem. The applications of the new monotonicity results are discussed.
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