 02401 COLIN de VERDIERE Yves
 THE LEVEL CROSSING PROBLEM IN SEMICLASSICAL ANALYSIS I. The symmetric
case
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Sep 25, 02

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Abstract. Our goal is to recover and extend
the difficult results of George Hagedorn
(1994)
on the
propagation of coherent states in the BornOppenheimer approximation
in the case of generic crossings of eigenvalues of the
(matrix valued) classical
Hamiltonian. This problem, going back to Landau and
Zener in the thirties, is often called the ``Mode Conversion Problem''
by physicists and occurs in many domains of physics.
We want to obtain a geometrical description of
the propagation of states
in the framework of semiclassical analysis
and WKBLagrangian states.
It turns out that, in a very beautiful (but not well known!) paper
published in 1993,
Peter Braam and Hans Duistermaat found that there is a formal normal
form for this problem. A formal normal form for the dispersion
relation were already founded by Arnold.
In our paper, we show, using Nelson's wave operators method,
that, in the hyperbolic case, their normal form
can be extended to a local normal form in the phase space.
Then, we extend
this classical
local normal form to the complete symbol, getting
a microlocal normal form, and derive from it a precise geometric
description of the semiclassical
propagation of states of a symmetric system of
pseudodifferential equations near a generic hyperbolic
codimension
3 singularity of the characteristic set
(defined by the so called ``dispersion relation'').
We describe in a sketchy way the elliptic case.
The complex
Hermitian case will be worked out in another paper.
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