Estimating complex eigenvalues of non-self-adjoint Schrödinger 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 Schrödinger operators $-\partial^2_x + x^2 + iγ f (x)$ with a complex potential. For $f$ of the special form$f (x) = 1/(1 + |x|^κ)$, it is shown using complex dilations that the real part of eigenvalues of the operator are larger than a constant times $|γ|^{2/(κ+2)}$.

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