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arXiv · 1505.05719

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

Patrick W. Dondl·Patrick Dorey·Frank Rösler
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Abstract

We are concerned with the non-normal Schrödinger operator $$ H=-Δ+V $$ 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 $δ>0$, $R>0$ there exists an $\varepsilon>0$ such that the $\varepsilon$-pseudospectrum $$ σ_\varepsilon(H)\subset \{z:\operatorname{Re}(z) \geq R\}\cup\bigcup_{λ\inσ(H)}\{z:|z-λ|<δ\}. $$ In particular, the unbounded part of the pseudospectrum escapes towards $+\infty$ as $\varepsilon$ decreases. Additionally, we give two examples of non-selfadjoint Schrödinger operators outside of our class and study their pseudospectra in more detail.

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Patrick W. Dondl, Patrick Dorey, Frank Rösler. 2017-01-08. A Bound on the Pseudospectrum of the Harmonic Oscillator with Imaginary Cubic Potential. https://doi.org/10.1093/amrx%2Fabw011

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