Abstract
We study non-elliptic quadratic differential operators. Quadratic differential operators are non-selfadjoint operators defined in the Weyl quantization by complex-valued quadratic symbols. When the real part of their Weyl symbols is a non-positive quadratic form, we point out the existence of a particular linear subspace in the phase space intrinsically associated to their Weyl symbols, called a singular space, such that when the singular space has a symplectic structure, the associated heat semigroup is smoothing in every direction of its symplectic orthogonal space. When the Weyl symbol of such an operator is elliptic on the singular space, this space is always symplectic and we prove that the spectrum of the operator is discrete and can be described as in the case of global ellipticity. We also describe the large time behavior of contraction semigroups generated by these operators.
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Acknowledgments
The research of the first author is supported in part by the National Science Foundation under grant DMS–0653275 and the Alfred P. Sloan Research Fellowship. He would also like to thank Joe Viola for a stimulating discussion. The second author is very grateful to Bernard Helffer and Francis Nier for their interest in his thesis work and for their very stimulating questions during the defense that we have tried to answer in the present work. We would also like to thank the referee for helpful comments leading to the improved presentation.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Hitrik, M., Pravda-Starov, K. Spectra and semigroup smoothing for non-elliptic quadratic operators. Math. Ann. 344, 801–846 (2009). https://doi.org/10.1007/s00208-008-0328-y
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DOI: https://doi.org/10.1007/s00208-008-0328-y