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 $1$D semi-groups generated by the Schrödinger operators $-\partial_{x}^{2}+x^{2 }+\im \gamma f(x)$ with a complex potential. For $f$ of the special form $f(x)=1/(1+|x|^{\kappa})$, it is shown using com-\break plex dilations that the real part of eigenvalues of the operator are larger than a constant times $ |\gamma|^{2/(\kappa+2)}$.

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Published 19 August 2011