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Three-body problem in \(d\)-dimensional space: ground state, (quasi)-exact-solvability. (English) Zbl 1384.81030

The 3-body problem is one of difficult for exact analytic treatment problems in celestial mechanics. An example of this problem, very actual and intriguing is the problem of motion of asteroids. If the asteroid is big enough, as is the case of Ceres, Pallada (Pallas), and few other the problem become difficult for analytic solving and, at the same time, important. Even the case of Icarus is important due to its very great ellipticity. The asteroid can enter into inner regions of the Solar system and can interact not only with Earth (which is vitally important for humankind) but also with smaller planets: Mars, Venus. Another interesting feature of this problem is resonances which can arises. Only the restricted 3-body problem, when the mass of 3-rd, body is small compared with masses of two other bodies has analytic solutions, which are known from research of such authors as E. A. Grebenikov, V. Demin, E. A. Aksenov, V.I. Arnold and A. N. Kolmogorov, while research of this problem begun with Charles Delaunay and G.W. Hill at the threeshold of XIX-th and XX-th century. Till now, mostly numeric methods were developed to estimate, e.g. the impact probability of asteroids with the Earth. Meanwhile, the analytic mathematicians and celestial mechanicians try to solve the problem as possible generally. The article under reviewing is just concerned to analytic treatment of the 3 -body problem in a \(d\)-dimensional space, where d-can be greater or less than 3. The authors try to formulate some general mathematic relations for arbitrary dimensional space-times, not only for a 3-dimensional one, we live in. Unfortunately nobody till now not claimed the physical necessity of introduction of more than 3-dimensional worlds, except for some visionaries as V. L. Ginzburg and V. P. Frolov for extragalactic astronomy. Only fantasts (and mathematicians, of course) except them discussed the occurence of more than 3 dimensions in the Macroworld.In view of the great importance of problem for future I agree that the problem can have in the future many citations. Concerning quantum aspects of the problem, it is necessary to separate the possible occurence of many dimensions in the microworld,close to Plankean values of mass, space and time, from the real 3 dimensional world , we live in. Meanwhile, authors study the “aspects of the quantum and classical dynamics of a 3-body system with equal masses, each body with d -degrees of freedom, with interaction depending only on mutual (relative) distances. The study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only”. What is the physical meaning of distance close to Plankean length? Would the Lagrangean methods still work at such very small distances? If yes, then the Ehrenfest’s theorem can be applied in order to make a transition from the microworld to obvious ours world. By the way, such works are actually widespread ( see, e.g. the works by Savas Dimopoulos and his disciples and colleagues: Asimina Arwanitaki, Nima Arkani-Hamed, Georgi Dvali and few other, which arises to the scientific schools of Frank Wilcsek and Leonard Susskind). It seems that a larger popularization of this area of investigation would be of use for many of mathematicians-beginners, physicists and interested. By the way, a very simple 2-body but Quantum Celestial mechanic in General Relativistic Schwarzschild field, solved by me in 1988–1990, has no any citations till now.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
70F15 Celestial mechanics
70F07 Three-body problems
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
17B63 Poisson algebras
81Q80 Special quantum systems, such as solvable systems
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References:

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