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

Abstract

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

Appl. Math. Res. Express (2016)

References in corpus (8)

• Making Sense of Non-Hermitian Hamiltonians
• Pseudospectra in non-Hermitian quantum mechanics
• PT-symmetric quantum mechanics
• PT-symmetric waveguides
• Differential operators admitting various rates of spectral projection growth
• Spectral projections of the complex cubic oscillator
• Semigroup Growth Bounds
• Estimation de résolvante et construction de quasimode près du bord du pseudospectre

Cited by in corpus (2)

• Resolvent estimates for one-dimensional Schrödinger operators with complex potentials
• Computation and Verification of Spectra for Non-Hermitian Systems