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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.

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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