Spectral projections of the complex cubic oscillator

Abstract

We prove the spectral instability of the complex cubic oscillator $-\frac{d^2}{dx^2}+ix^3+iαx$ for non-negative values of the parameter $α$, by getting the exponential growth rate of $\|Π_n(α)\|$, where $Π_n(α)$ is the spectral projection associated with the $n$-th eigenvalue of the operator. More precisely, we show that for all non-negative $α$ \[ \lim\limits_{n\to+\infty}\frac{1}{n}\log\|Π_n(α)\| = \fracπ{\sqrt{3}}. \]