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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 $|\gamma|^{2/(\kappa+2)}$.

Estimating Complex Eigenvalues of Non-Self Adjoint Schrödinger Operators via Complex Dilations