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A complete study of the pseudo-spectrum for the rotated harmonic oscillator. (English) Zbl 1106.34060
The pseudo-spectrum of an operator describes the behavior of the spectrum to small perturbations: the $$\varepsilon$$ pseudo-spectrum of a closed operator $$A$$ is the set of complex numbers $$z$$ such that there exists a perturbation $$\Delta A$$ of $$A$$ having norm $$\leq \varepsilon$$ and such that $$z$$ belongs to the spectrum of $$A+ \Delta A$$. This notion is especially useful in the case of nonselfadjoint operators. The study of pseudo-spectra was developed in the last 20 years and was partially motivated by its applications to the numerical computation of the eigenvalues of an operator.
The present paper contains a proof of a conjecture due to L. S. Boulton [J. Oper. Theory, 47, 413-429 (2002; Zbl 1034.34099)] concerning the shape of the $$\varepsilon$$ pseudo-spectrum of the rotated harmonic oscillator $$H_{c}=-{d^{2}\over dx^{2}}+cx^{2}$$, $$\text{Re\,}c,\text{Im\,}c>0$$. The proof is based on notions and methods from the theory of semiclassical analysis of pseudo-differential operators (see e. g. [N. Dencker, J. Sjøstrand and M. Zworski, Commun. Pure Appl. Math., 57, 384-415 (2004; Zbl 1054.35035)]), but is more elementary. Its core is an explicit estimation on the semi-classical pseudo-spectrum of the rotated harmonic oscillator, which uses some localization scheme in the frequency variable.

##### MSC:
 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 47A10 Spectrum, resolvent 35S99 Pseudodifferential operators and other generalizations of partial differential operators
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