  The phenomenon "hypo-coercivity," i.e., the increased rate of contraction for
a semi-group upon adding a large skew-adjoint part to the generator, is
considered for 1D semigroups generated by the Schr\"odinger operators
$-\partial^2_x + x^2 + i{\gamma} f (x)$ with a complex potential. For $f$ of
the special form$ f (x) = 1/(1 + |x|^\kappa)$, it is shown using complex
dilations that the real part of eigenvalues of the operator are larger than a
constant times $|\gamma|^{2/(\kappa+2)}$.
