Semiclassical Schr\"odinger operators with purely imaginary potential

We consider Schrödinger operators with purely imaginary potential $P = - h^{2} \Delta + i V ( x )$ on a bounded domain.

Assuming that near its critical points the potential $V$ can be approximated by an homogeneous polynomial, we show that in the limit $h \to 0$ the leftmost eigenvalues of $P$ are asymptotically given by the local model associated to the most degenerated critical points of $V$.

We give applications of this result to the associated evolution problem including shear flows in fluid mechanics.