Spectral projections and resolvent bounds for partially elliptic quadratic differential operators

Viola, Joe

doi:10.1007/s11868-013-0066-0

Cited by 35 publications

(48 citation statements)

References 26 publications

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“…Theorem 1.1 applies with the matrix M and the weight Φ(z) = 1 2 |z| 2 for z ∈ C 2 (see, e.g., [32,Ex. 2.7]), and because A + = A − , we are in a situation where (1.9) holds.…”

Section: 22mentioning

confidence: 99%

On weak and strong solution operators for evolution equations coming from quadratic operators

Aleman

Viola

2018

J. Spectr. Theory

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We identify, through a change of variables, solution operators for evolution equations with generators given by certain simple first-order differential operators acting on Fock spaces. This analysis applies, through unitary equivalence, to a broad class of supersymmetric quadratic multiplicationdifferentiation operators acting on L 2 (R n ) which includes the elliptic and weakly elliptic quadratic operators. We demonstrate a variety of sharp results on boundedness, decay, and return to equilibrium for these solution operators, connecting the short-time behavior with the range of the symbol and the long-time behavior with the eigenvalues of their generators. This is particularly striking when it allows for the definition of solution operators which are compact and regularizing for large times for certain operators whose spectrum is the entire complex plane.

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“…Theorem 1.1 applies with the matrix M and the weight Φ(z) = 1 2 |z| 2 for z ∈ C 2 (see, e.g., [32,Ex. 2.7]), and because A + = A − , we are in a situation where (1.9) holds.…”

Section: 22mentioning

confidence: 99%

On weak and strong solution operators for evolution equations coming from quadratic operators

Aleman

Viola

2018

J. Spectr. Theory

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“…which are respectively the Hamilton maps of the quadratic forms Re and Im . As pointed out in [6,15,17,22], the singular space is playing a basic role in understanding the spectral and hypoelliptic properties of non-elliptic quadratic operators, as well as the spectral and pseudospectral properties of certain classes of degenerate doubly characteristic pseudodifferential operators [7,8,21].…”

Section: Introductionmentioning

confidence: 99%

Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians

Pravda-Starov

Rodino

Wahlberg

2017

Mathematische Nachrichten

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Abstract. We study propagation of the Gabor wave front set for a Schrödinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with non-negative real part. We point out that the singular space associated to the quadratic form plays a crucial role for the understanding of this propagation. We show that the Gabor singularities of the solution to the equation for positive times are always contained in the singular space, and that they propagate in this set along the flow of the Hamilton vector field associated to the imaginary part of the quadratic form. As an application we obtain for the heat equation a sufficient condition on the Gabor wave front set of the initial datum tempered distribution that implies regularization to Schwartz regularity for positive times.

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“…As pointed out in [10,24,27,30], the singular space plays a basic role in understanding the spectral and hypoelliptic properties of non-elliptic quadratic operators, as well as the spectral and pseudospectral properties of certain classes of degenerate doubly characteristic pseudodifferential operators [11,12,29]. Some applications of these results to the study of degenerate hypoelliptic Ornstein-Uhlenbeck operators and degenerate hypoelliptic Fokker-Planck operators are also given in [25].…”

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confidence: 97%

Short-time asymptotics of the regularizing effect for semigroups generated by quadratic operators

Hitrik

Pravda-Starov

Viola

2017

Bulletin des Sciences Mathématiques

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46 pages. arXiv admin note: text overlap with arXiv:1411.6223International audienceWe study accretive quadratic operators with zero singular spaces. These degenerate non-selfadjoint differential operators are known to be hypoelliptic and to generate contraction semigroups which are smoothing in the Schwartz space for any positive time. In this work, we study the short-time asymptotics of the regularizing effect induced by these semigroups. We show that these short-time asymptotics of the regularizing effect depend on the directions of the phase space, and that this dependence can be nicely understood through the structure of the singular space. As a byproduct of these results, we derive sharp subelliptic estimates for accretive quadratic operators with zero singular spaces pointing out that the loss of derivatives with respect to the elliptic case also depends on the phase space directions according to the structure of the singular space. Some applications of these results are then given to the study of degenerate hypoelliptic Ornstein-Uhlenbeck operators and degenerate hypoelliptic Fokker-Planck operators

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Spectral projections and resolvent bounds for partially elliptic quadratic differential operators

Cited by 35 publications

References 26 publications

On weak and strong solution operators for evolution equations coming from quadratic operators

On weak and strong solution operators for evolution equations coming from quadratic operators

Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians

Short-time asymptotics of the regularizing effect for semigroups generated by quadratic operators

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