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Non-trapping condition for semiclassical Schrödinger operators with matrix-valued potentials. (English) Zbl 1067.81037

Summary: We consider semiclassical Schrödinger 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 a certain special condition on the matricial structure of the potential at the eigenvalues crossing and a certain structure condition at infinity, we prove that the boundary values of the resolvent at energy \(\lambda\), as bounded operators on suitable weighted spaces, are \(O(h^{-1})\) if and only if \(\lambda\) is a non-trapping energy for all the Hamilton flows generated by the eigenvalues of the operator’s symbol.

MSC:

81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35P99 Spectral theory and eigenvalue problems for partial differential equations
47A10 Spectrum, resolvent
47F05 General theory of partial differential operators
47N50 Applications of operator theory in the physical sciences
81Q15 Perturbation theories for operators and differential equations in quantum theory
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81R30 Coherent states
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