- 04-91 Thierry Jecko
- Non-trapping condition for semiclassical Schr dinger operators with matrix-valued potentials.
Mar 23, 04
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Abstract. We consider semiclassical Schr dinger operators with matrix-valued,
long-range, smooth potential, for which different eigenvalues may cross
on a codimension one submanifold. We denote by h the semiclassical
parameter and we consider energies above the bottom of the essential spectrum. Under some invariance condition on the matricial structure of the potential near the eigenvalues crossing and some structure condition at infinity, we prove that the boundary values of the resolvent at energy lambda, as bounded operators on suitable weighted spaces, are O(1/h) if and only if lambda is a non-trapping energy for all the Hamilton flows
generated by the eigenvalues of the operator's symbol.