  We study the pseudospectrum of a class of non-selfadjoint differential
operators. Our work consists in a detailed study of the microlocal properties,
which rule the spectral stability or instability phenomena appearing under
small perturbations for elliptic quadratic differential operators. The class of
elliptic quadratic differential operators stands for the class of operators
defined in the Weyl quantization by complex-valued elliptic quadratic symbols.
We establish in this paper a simple necessary and sufficient condition on the
Weyl symbol of these operators, which ensures the stability of their spectra.
When this condition is violated, we prove that it occurs some strong spectral
instabilities for the high energies of these operators, in some regions which
can be far away from their spectra. We give a precise geometrical description
of them, which explains the results obtained for these operators in some
numerical simulations giving the computation of false eigenvalues far from
their spectra by algorithms for eigenvalues computing.
