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A trigonometric sum relevant to the nonrelativistic theory of atoms. (English) Zbl 0827.11079

The authors extend the method of Van der Corput (devised to study trigonometric sums appearing in different number theoretic problems) for exponential sums in order to study an oscillatory term which appears in the quantum theory of large atoms. Of special interest in this context is the lattice point problem, which consists in estimating the number of lattice points inside a large convex region.
According to the method, after Poisson resummation the relevant expression becomes a sum of Fourier integrals. Each integral can be expanded by means of the stationary-phase method to become a sum of complex exponentials with real amplitudes. A crucial ingredient in the analysis is the non-degeneracy of the Thomas-Fermi potential.
An interpretation of the result in terms of classical dynamics is given. Finally, sharp asymptotic upper and lower bounds for the oscillations are obtained too.

MSC:

11Z05 Miscellaneous applications of number theory
81Q99 General mathematical topics and methods in quantum theory
11L03 Trigonometric and exponential sums (general theory)
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