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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.

MSC:

34L05 General spectral theory of ordinary differential operators
47E05 General theory of ordinary differential operators
34D10 Perturbations of ordinary differential equations
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