Another Application of Dilation Analytic Method for Complex Lieb--Thirring Type Estimates

We consider non-self-adjoint Schr\"{o}dinger operators $H_{{\rm c}}=-\Delta+V_{{\rm c}}$ (resp. $H_{{\rm r}}=-\Delta+V_{{\rm r}}$) acting in $L^2(\mathbb R^d)$, $d\ge 1$, with dilation analytic complex (resp. real) potentials. We were able to find out perhaps a new application of dilation analytic method in \cite{So1} (N. Someyama, "Number of Eigenvalues of Non-self-adjoint Schr\"{o}dinger Operators with Dilation Analytic Complex Potentials," Reports on Mathematical Physics, Volume 83, Issue 2, pp.163-174 (2019).). We give a Lieb--Thirring type estimate on resonance eigenvalues of $H_{{\rm c}}$ in the open complex sector and that on embedded eigenvalues of $H_{{\rm r}}$ in the same way as \cite{So1}. To achieve that, we derive Lieb--Thirring type inequalities for isolated eigenvalues of $H$ on several complex subplanes.