Semiclassical asymptotics, gauge fields, and quantum chaos. (English) Zbl 0679.58046
The paper continues the authors’ study [Commun. Math. Phys. 92, 555-594 (1984; Zbl 0534.58028)] of the spectral behavior in the limit \(\hslash \to 0\) for nonrelativistic Schrödinger operators. Starting from a nonrelativistic gauge field given as connection in a principal G-bundle over a Riemannian manifold M, the paper investigates the semiclassical behavior of Schrödinger operators associated with particles in the gauge field. These particles are (as classical particles) canonically described by their extended configuration space \(\pi^*(P\times ad {\mathfrak g})\) (\({\mathfrak g}\) is the Lie algebra of G), which is a Poisson manifold.
After finding a suitable form for the Hamiltonian operator as a function of \(\hslash\), Fourier integral operator techniques are used intensively to study the asymptotic behavior. The symbols are naturally defined on the extended configuration space (in the paper called WGS bundle) of associated classical particles. The results are used to study smoothed out Chern forms and to describe how ergodicity and chaotic behavior of the flow of these particles is reflected by the spectral behavior of the Hamiltonian operators.
After finding a suitable form for the Hamiltonian operator as a function of \(\hslash\), Fourier integral operator techniques are used intensively to study the asymptotic behavior. The symbols are naturally defined on the extended configuration space (in the paper called WGS bundle) of associated classical particles. The results are used to study smoothed out Chern forms and to describe how ergodicity and chaotic behavior of the flow of these particles is reflected by the spectral behavior of the Hamiltonian operators.
Reviewer: Ch.Günther
MSC:
| 58J40 | Pseudodifferential and Fourier integral operators on manifolds |
| 81T08 | Constructive quantum field theory |
| 47Gxx | Integral, integro-differential, and pseudodifferential operators |
Keywords:
semiclassical limit; asymptotics; Fourier integral operators; nonrelativistic Schrödinger operators; nonrelativistic gauge fieldCitations:
Zbl 0534.58028References:
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