A Bound on the Pseudospectrum of the Harmonic Oscillator with Imaginary Cubic Potential

We are concerned with the non-normal Schr"odinger operator $$H=-\mathrm{\Delta }+V$$ on $L^2({\mathbb{R}}^n)$, where $V\in W^{1,\infty}_{$\text{loc}$}($\mathbb{R}$^n)$and$\operatorname{Re} (V(x))\ge c|x|^2-d$forsome$c,d>0$. The spectrum of this operator is discrete and contained in the positive half plane. In general, the $\epsilon$-pseudospectrum of $H$ will have an unbounded component for any $\epsilon >0$ and thus will not approximate the spectrum in a global sense. By exploiting the fact that the semigroup $e^{-tH}$ is immediately compact, we show a complementary result, namely that for every $\delta >0$, $R>0$ there exists an $\epsilon >0$ such that the $\epsilon$-pseudospectrum $${\sigma }_{\epsilon }(H)\subset \{z:\mathrm{Re}(z)\ge R\}\cup \bigcup _{\lambda \in \sigma (H)}\{z:\mathrm{\mid }z-\lambda \mathrm{\mid }<\delta \}\mathrm{.}$$ In particular, the unbounded part of the pseudospectrum escapes towards $+\mathrm{\infty }$ as $\epsilon$ decreases. Additionally, we give two examples of non-selfadjoint Schr"odinger operators outside of our class and study their pseudospectra in more detail.