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Monotonicity of Quantum Ground State Energies: Bosonic Atoms and Stars

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Abstract

The N-dependence of the non-relativistic bosonic ground state energy ℰB(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.

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  1. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854, USA

    Michael K.-H. Kiessling

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Kiessling, M.KH. Monotonicity of Quantum Ground State Energies: Bosonic Atoms and Stars. J Stat Phys 137, 1063–1078 (2009). https://doi.org/10.1007/s10955-009-9843-9

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