Journal of Spectral Theory

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Volume 4, Issue 2, 2014, pp. 349–364
DOI: 10.4171/JST/72

Published online: 2014-07-13

Spectral instability for even non-selfadjoint anharmonic oscillators

Raphaël Henry[1]

(1) Université Paris-Sud, Orsay, France

We study the instability of the spectrum for a class of non-selfadjoint anharmonic oscillators, estimating the behavior of the instability indices (\emph{i. e.} the norm of spectral projections) associated with the large eigenvalues of these oscillators. More precisely, we consider the operators \[ \CA(2k,\theta) = -\frac{d^2}{dx^2}+e^{i\theta}x^{2k} \] defined on $L^2(\mathbb{R})$, with $k\geq1$ and $|\theta|<(k+1)\pi/2k$. We get asymptotic expansions for the instability indices, extending the results of \cite{Dav2} and \cite{DavKui}.

Keywords: Non-selfadjoint operators, complex WKB method, asymptotic expansions, completeness of eigenfunctions

Henry Raphaël: Spectral instability for even non-selfadjoint anharmonic oscillators. J. Spectr. Theory 4 (2014), 349-364. doi: 10.4171/JST/72