Abstract
We prove that smooth Wigner-Weyl spectral sums at an energy level E exhibit Airy scaling asymptotics across the classical energy surface ΣE . This was proved earlier by the authors for the isotropic harmonic oscillator and the proof is extended in this article to all quantum Hamiltonians −ℏ 2Δ + V where V is a confining potential with at most quadratic growth at infinity. The main tools are the Herman-Kluk initial value parametrix for the propagator and the Chester-Friedman-Ursell normal form for complex phases with a one-dimensional cubic degeneracy. This gives a rigorous account of Airy scaling asymptotics of spectral Wigner distributions of Berry, Ozorio de Almeida and other physicists.
Original language | English (US) |
---|---|
Article number | 414003 |
Journal | Journal of Physics A: Mathematical and Theoretical |
Volume | 55 |
Issue number | 41 |
DOIs | |
State | Published - Oct 14 2022 |
Funding
SZ was partially supported by NSF Grant DMS-1810747. BH was funded by NSF CAREER Grant DMS-2143754 as well as NSF Grants DMS-1855684, DMS-2133806 and an ONR MURI on Foundations of Deep Learning.
Keywords
- Airy
- Wigner-Weyl
- asymptotics
- spectral
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Statistics and Probability
- Modeling and Simulation
- Mathematical Physics
- General Physics and Astronomy