On the pseudospectrum of elliptic quadratic differential operators

We study the pseudospectrum of a class of nonselfadjoint differential operators. Our work consists of a microlocal study of the properties that 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 article 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 strong spectral instabilities occur for the high energies of these operators in some regions that can be far away from their spectra