The Hartree limit of Born's ensemble for the ground state of a bosonic atom or ion (Q2872269)

Mathematical physics has been a very rich and interesting field of study and research, particularly in the past several decades. The power and rigour of mathematics as well as the wealth of factual materials possessed by physics could enhance and enrich one's actual understanding of nature multifold, as clearly observed in recent times. The paper under review is an exemplary specimen of highly sophisticated methods and techniques used to bring out quite valuable results concerning bosonic atomic or ionic and certain related aspects of matter.NEWLINENEWLINE The first section of the article is a detailed introduction drawing attention to some salient features of the work like the Hartree approximation to the ground state of a non-relativistic atom/ion with bosonic ``electrons''. Precise statements of results are given in the second section covering the \(N\)-body variational principle, the Hartree approximation, Born's statistical ensemble and two good theorems. The next section contains the proof of the first theorem whose ensemble reformulation is followed. A third theorem and its proof are presented there. The proof of the second theorem is also furnished. Finally the author enters the probabilistic meaning of Born's ensemble for the rescaled ground state of the initially chosen Hamiltonian. It is claimed that the notion of typicality newly introduced stands in terms by which everything is said about the second theorem through the probability idea can be rephrased.NEWLINENEWLINE The fourth section contains a number of important concluding remarks concerning the novel approach to the Hartree limit of bosonic Born-Oppenheimer atomes/ions, with possible generalization of the researcher's new techniques, hinted at.NEWLINENEWLINE The 21-page illustrious science writing has 10 remarks, 3 conjectures, 3 corollaries, 4 propositions, 2 lemmas, 72 equations and 77 ``references''.