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

We are concerned with the non-normal Schr\"odinger operator $$ H=-\Delta+V $$ on $ L^2(\mathbb R^n)$, where $V\in W^{1,\infty}_{\text{loc}}(\mathbb{R}^n)$ and $\operatorname{Re} (V(x))\ge c|x|^2-d$ for some $c,d>0$. The spectrum of this operator is discrete and contained in the positive half plane. In general, the $\varepsilon$-pseudospectrum of $H$ will have an unbounded component for any $\varepsilon>0$ 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 $\varepsilon>0$ such that the $\varepsilon$-pseudospectrum $$ \sigma_\varepsilon(H)\subset \{z:\operatorname{Re}(z) \geq R\}\cup\bigcup_{\lambda\in\sigma(H)}\{z:|z-\lambda|<\delta \}. $$ In particular, the unbounded part of the pseudospectrum escapes towards $+\infty$ as $\varepsilon$ decreases. Additionally, we give two examples of non-selfadjoint Schr\"odinger operators outside of our class and study their pseudospectra in more detail.