Estimating complex eigenvalues of non-self-adjoint Schr\"odinger operators via complex dilations

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 $-{\mathrm{\partial }}_x^2+x^2+i\gamma f(x)$ with a complex potential. For $f$ of the special form$f(x)=1\mathrm{/}(1+\mathrm{\mid }x{\mathrm{\mid }}^{\kappa })$, it is shown using complex dilations that the real part of eigenvalues of the operator are larger than a constant times $\mathrm{\mid }\gamma {\mathrm{\mid }}^{2\mathrm{/}(\kappa +2)}$.