Spectral Properties of Non-self-adjoint Operators in the Semi-classical Regime

We give a spectral description of the semi-classical Schrödinger operator with a piecewise-linear, complex-valued potential. Moreover, using these results, we give an example to show how an arbitrarily small fixed perturbation of a non-self-adjoint operator can completely change the asymptotic spectrum of the operator in the semi-classical limit.