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Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations

Part of the book series: Pseudo-Differential Operators ((PDO,volume 14))

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Abstract

Non-self-adjoint operators is an old, sophisticated and highly developed subject. See for instance Carleman for an early result on Weyl type asymptotics for the real parts of the large eigenvalues of operators that are close to self-adjoint ones, with later results by Markus and Matseev in the same direction. (See the classical works Weyl, Avakumović , Hörmander for the asymptotics of large eigenvalues of elliptic self-adjoint operators and Robert and Dimassi and Sjöstrand for corresponding results in the semi-classical case, not to mention numerous deep and sophisticated results by Ivrii and others.) Abstract theory with the machinery of s-numbers can be found in the book of Gohberg and Krein. Other quite classical results concern upper bounds on the number of eigenvalues in various regions of the complex plane and questions about completeness of the set of all generalized eigenvectors.

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Notes

  1. 1.

    At first, the formula appeared more complicated, depending on the method of proof, and the simpler form was pointed out to me by E. Amar-Servat.

  2. 2.

    Including several articles of the author, often with minor changes.

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Acknowledgements

Discussions with colleagues and coworkers have been a very important basis for this work. I am grateful to W. Bordeaux Montrieux, M. Hager, M. Hitrik, B. Helffer, K. Pravda-Starov, J. Viola, M. Vogel, X. P. Wang, M. Zworski and many others. We also thank the two referees for a very careful work and most useful comments.

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    Johannes Sjöstrand

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Sjöstrand, J. (2019). Introduction. In: Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations. Pseudo-Differential Operators, vol 14. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-10819-9_1

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