|
This article is cited in 4 scientific papers (total in 4 papers)
Circle problem and the spectrum of the Laplace operator on closed 2-manifolds
D. A. Popov Lomonosov Moscow State University, Belozerskii Research Institute for Physical and Chemical Biology
Abstract:
In this survey the circle problem is treated in the broad sense, as the problem of the asymptotic properties of the quantity $P(x)$, the remainder term in the circle problem. A survey of recent results in this direction is presented. The main focus is on the behaviour of $P(x)$ on short intervals. Several conjectures on the local behaviour of $P(x)$ which lead to a solution of the circle problem are presented. A strong universality conjecture is stated which links the behaviour of $P(x)$ with the behaviour of the second term in Weyl's formula for the Laplace operator on a closed Riemannian 2-manifold with integrable geodesic flow.
Bibliography: 43 titles.
Keywords:
circle problem, Voronoi's formula, short intervals, quantum chaos, universality conjecture.
Received: 01.12.2018
Citation:
D. A. Popov, “Circle problem and the spectrum of the Laplace operator on closed 2-manifolds”, Uspekhi Mat. Nauk, 74:5(449) (2019), 145–162; Russian Math. Surveys, 74:5 (2019), 909–925
Linking options:
https://www.mathnet.ru/eng/rm9911https://doi.org/10.1070/RM9911 https://www.mathnet.ru/eng/rm/v74/i5/p145
|
Statistics & downloads: |
Abstract page: | 367 | Russian version PDF: | 93 | English version PDF: | 46 | References: | 46 | First page: | 26 |
|