100 years of Weyl’s law. (English) Zbl 1358.35075
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 a 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.
MathOverflow Questions:
Some question about the spectral function of Laplace operator on \(\mathbb{R}^n\)MSC:
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35J10 | Schrödinger operator, Schrödinger equation |
References:
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