Abstract
We study resolvents and spectral projections for quadratic differential operators under an assumption of partial ellipticity. We establish exponential-type resolvent bounds for these operators, including Kramers–Fokker–Planck operators with quadratic potentials. For the norms of spectral projections for these operators, we obtain complete asymptotic expansions in dimension one, and for arbitrary dimension, we obtain exponential upper bounds and the rate of exponential growth in a generic situation. We furthermore obtain a complete characterization of those operators with orthogonal spectral projections onto the ground state.
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Acknowledgments
The author would like to thank Michael Hitrik for helpful discussions at the outset of this work. He would also like to thank Frédéric Hérau for helpful suggestions and Fredrik Andersson for a valuable discussion contributing to the proof of Proposition 4.1. Finally, the author would like to thank the referee for a careful reading and many helpful corrections and suggestions.
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Viola, J. Spectral projections and resolvent bounds for partially elliptic quadratic differential operators. J. Pseudo-Differ. Oper. Appl. 4, 145–221 (2013). https://doi.org/10.1007/s11868-013-0066-0
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DOI: https://doi.org/10.1007/s11868-013-0066-0
Keywords
- Non-selfadjoint operator
- Resolvent estimate
- Spectral projections
- Quadratic differential operator
- FBI-Bargmann transform