Mathematical Research Letters

Volume 18 (2011)

Number 4

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

Pages: 755 – 765



Jeffrey H. Schenker (Department of Mathematics, Michigan State University, East Lansing, MI 48823, USA)


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)}$.

Published 19 August 2011