Semiclassical spectra of gauge fields. (English) Zbl 0772.58066
The authors study the asymptotic behavior of the eigenvalues of the Schrödinger operator with a vector potential on a compact manifold, for Planck’s constant tending to zero.
Given a principal compact \(G\)-bundle \(P\) over a Riemannian manifold \(M\), the spectral behavior is investigated in terms of the joint spectrum of commuting operators on \(P\), stipulating geometric assumptions which lead to studying functions of operators of real principal type.
Estimates in terms of periodic trajectories of Wong’s flow, which are uniform in the “charge” parameter, are obtained; a family of examples involving particularly \(G=U(2)\) which illustrate some types of classical and non-classical asymptotics, and a possible generalization developed by means of a given Higgs field are also enclosed.
Given a principal compact \(G\)-bundle \(P\) over a Riemannian manifold \(M\), the spectral behavior is investigated in terms of the joint spectrum of commuting operators on \(P\), stipulating geometric assumptions which lead to studying functions of operators of real principal type.
Estimates in terms of periodic trajectories of Wong’s flow, which are uniform in the “charge” parameter, are obtained; a family of examples involving particularly \(G=U(2)\) which illustrate some types of classical and non-classical asymptotics, and a possible generalization developed by means of a given Higgs field are also enclosed.
Reviewer: V.Balan (Bucureşti)
MSC:
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 35P05 | General topics in linear spectral theory for PDEs |
| 58J40 | Pseudodifferential and Fourier integral operators on manifolds |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
Keywords:
Fourier analysis; radiality; eigenvalues; Schrödinger operator; operators of real principal type; Wong’s flowReferences:
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