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Semiclassical spectral instability for non-self-adjoint operators. I: A model. (Instabilité spectrale semiclassique pour des opérateurs non-autoadjoints. I: un modèle.) (French) Zbl 1131.34057
A simple 1-dimensional model problem is discussed and explored. Let \(\partial\) denote 1-dimensional differentiation, then \(-\mathrm{i}\,\partial\) is considered as a selfadjoint operator on the interval \(\left[0,2\pi\right]\) with periodicity boundary conditions. Perturbations associated with this operator of the form \(P=-\mathrm{i}\, h\,\partial+g\left(m\right)\) are considered, where \(g\left(m\right)\) indicates the multiplication operator \(\left(g\left(m\right)f\right)\left(x\right)=g\left(x\right)\, f\left(x\right)\), \(x\in\left[0,2\pi\right]\), for an analytic complex-valued function \(g\) with \(g^{\prime}\) having a nonvanishing imaginary part and \(h\in]0,1]\) is a parameter. Whereas the spectrum of \(-\mathrm{i}\, h\,\partial\) is simply \(\frac{2\pi}{h}\left[\mathbb{Z}\right]\), it is shown that for the perturbed case the eigenvalues of \(P\) are distributed within the semi-classical “pseudo”-spectrum (given as the closure of the range of the semi-classical symbol \(\left(x,\xi\right)\mapsto h\xi+g\left(x\right)\)) according to a two-dimensional Weyl law.

34L05 General spectral theory of ordinary differential operators
47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX)
34D10 Perturbations of ordinary differential equations
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