Spectra and semigroup smoothing for non-elliptic quadratic operators
By Michael Hitrik, K. Pravda-Starov. 2007
72 citations recorded by the source corpus.
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.
Subfields: Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/80a8fcef9dec8b8462447dfd1e819bf9314af837
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Citations in the corpus
- Spectral estimates for finite combinations of Hermite functions and null-controllability of hypoelliptic quadratic equations (2018, background): “As pointed out in [23, 26, 27, 38, 39, 40, 48], the notion of singular space plays a basic role in the understanding of the spectral and hypoelliptic properties of the (possibly) nonelliptic quadratic operator q(x,Dx), as well as the spectral and pseudospectral properties of certain classes of degen”
- Null-controllability of hypoelliptic quadratic differential equations (2016, unknown): “The notion of singular space was introduced in [29] by Hitrik and the second author by pointing out the existence of a particular vector subspace in the phase space S ⊂ R, which is intrinsically associated with a quadratic symbol q.”