## Duke Mathematical Journal

### On the pseudospectrum of elliptic quadratic differential operators

Karel Pravda-Starov

#### Abstract

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

#### Article information

Source
Duke Math. J., Volume 145, Number 2 (2008), 249-279.

Dates
First available in Project Euclid: 20 October 2008

https://projecteuclid.org/euclid.dmj/1224508837

Digital Object Identifier
doi:10.1215/00127094-2008-051

Mathematical Reviews number (MathSciNet)
MR2449947

Zentralblatt MATH identifier
1157.35129

Subjects
Primary: 35S05: Pseudodifferential operators
Secondary: 35P05: General topics in linear spectral theory

#### Citation

Pravda-Starov, Karel. On the pseudospectrum of elliptic quadratic differential operators. Duke Math. J. 145 (2008), no. 2, 249--279. doi:10.1215/00127094-2008-051. https://projecteuclid.org/euclid.dmj/1224508837

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