Resonance free domain for a system of Schrödinger operators with energy-level crossings

Abstract

We consider a 2 × 2 system of 1D semiclassical differential operators with two Schrödinger operators in the diagonal part and small interactions of order h ν in the off-diagonal part, where h is a semiclassical parameter and ν is a constant larger than 1/2. We study the absence of resonance near a non-trapping energy for both Schrödinger operators in the presence of crossings of their potentials. The width of resonances is estimated from below by M h log(1/h) and the coefficient M is given in terms of the directed cycles of the generalized bicharacteristics induced by two Hamiltonians.