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

TL;DR

This paper establishes a bound on the pseudospectrum of a class of non-normal Schrödinger operators with quadratic growth potentials, showing that the unbounded pseudospectrum component moves towards infinity as the approximation parameter decreases.

Contribution

It proves a new bound on the pseudospectrum of non-selfadjoint Schrödinger operators with quadratic potentials, highlighting the asymptotic behavior of the unbounded component.

Findings

Unbounded pseudospectrum component escapes to infinity as ε decreases.

The spectrum is discrete and contained in the positive half-plane.

Semigroup e^{-tH} is immediately compact, enabling the pseudospectrum bound.

Abstract

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.