Spectral projections of the complex cubic oscillator

Abstract

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

References in corpus (1)

• Making Sense of Non-Hermitian Hamiltonians

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