100 years of Weyl’s law

We discuss the asymptotics of the eigenvalue counting function for partial differential operators and related expressions paying the most attention to the sharp asymptotics. We consider Weyl asymptotics, asymptotics with Weyl principal parts and correction terms and asymptotics with non-Weyl princip...

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Bibliographic Details
Published inBulletin of mathematical sciences Vol. 6; no. 3; pp. 379 - 452
Main Author Ivrii, Victor
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.10.2016
World Scientific Publishing Co. Pte., Ltd
World Scientific Publishing
Subjects
Mathematics
Mathematics and Statistics
Eigenvalue Counting Function
Asymptotic Pointwise
Thomas-Fermi Theory
Microhyperbolicity
Remainder Estimate
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Summary:We discuss the asymptotics of the eigenvalue counting function for partial differential operators and related expressions paying the most attention to the sharp asymptotics. We consider Weyl asymptotics, asymptotics with Weyl principal parts and correction terms and asymptotics with non-Weyl principal parts. Semiclassical microlocal analysis, propagation of singularities and related dynamics play crucial role. We start from the general theory, then consider Schrödinger and Dirac operators with the strong magnetic field and, finally, applications to the asymptotics of the ground state energy of heavy atoms and molecules with or without a magnetic field.
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ISSN:1664-3607
1664-3615
DOI:10.1007/s13373-016-0089-y