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