Quantum limits for subelliptic operators

Keywords:

• Semi-classical analysis in sub-Riemannian and sub-elliptic contexts

• Micro-local analysis on filtered manifolds

• Quantum ergodicity for sub-elliptic operators

Abstract:

Semi-classical analysis aims at understanding mathematically the transition from quantum to classical mechanics. A key question is to analyse the solutions to Schrödinger's equation. Almost all current and previous research has been focused on this equation involving elliptic operators; this is a convenient mathematical hypothesis but it excludes more general operators known as sub-elliptic. Sub-elliptic (and non-elliptic) operators appear naturally in non-Euclidean geometry with degenerate directions such as contact geometry, thereby having significant applications in mechanics, optics, thermodynamics and control theory. Our proposed research investigates the semi-classical analysis of sub-elliptic operators.

The team in alphabetical order:

Collaborators:

The publications in reverse chronological order:

• Veronique Fischer and Francesca Tripaldi.
  Subcomplexes on filtered manifolds. Preprint.

• Steven Flynn.
  The sub-Riemannian X-ray Transform on H-type groups: Fourier-Slice Theorems and Injectivity sets. Preprint.

• Søren Mikkelsen.
  Sharp semiclassical spectral asymptotics for Schrödinger operators with non-smooth potentials. Preprint.

• Søren Mikkelsen.
  Sharp semiclassical spectral asymptotics for local magnetic Schrödinger operators on Rd without full regularity. Preprint. 

• Lino Benedetto, Clotilde Fermanian-Kammerer and Veronique Fischer.
  Quantization on Groups and Garding inequality. Preprint

• Steven Flynn. 

Singular value decomposition for the X-ray transform on the reduced Heisenberg group, and a two-radius theorem. Preprint 

• Clotilde Fermanian-Kammerer, Veronique Fischer and Steven Flynn. 

Some remarks on semi-classical analysis on two-step nilmanifolds.

To appear in the proceedings of IQM22, Milano, Italy. Preprint 

• Veronique Fischer and Francesca Tripaldi. 

An alternative construction to the Rumin complex on homogeneous nilpotent Lie groups.  

Published in Advances in Mathematics. Preprint 

• Clotilde Fermanian-Kammerer, Veronique Fischer and Steven Flynn. 

Geometric invariance of the semi-classical calculus on nilpotent graded Lie groups. 

Published in Journal of Geometric Analysis. Preprint 

• Veronique Fischer. 

Asymptotics and zeta functions on nilmanifolds.

Published in Journal des Mathématiques Pures et Appliquées. Preprint 

• Veronique Fischer. 

Towards semi-classical analysis for sub-elliptic operators. 

Published in Bruno Pini Mathematical Analysis Seminar 2021. Preprint

• Clotilde Fermanian-Kammerer and Cyril Letrouit.

Observability and controllability for the Schrödinger equation on quotients of groups of Heisenberg type. 

Published in Journal de l'École Polytechnique. Preprint 

• Veronique Fischer. 

Semiclassical analysis on compact nilmanifolds. Preprint 

• Clotilde Fermanian-Kammerer and Veronique Fischer. 

Quantum evolution and sub-laplacian operators on groups of Heisenberg type. 

Published in Journal of Spectral Theory. Preprint 

• Clotilde Fermanian-Kammerer and Veronique Fischer. 

Semi-classical analysis on H-type groups. 

Published in Science China Mathematics.  Preprint 

• Clotilde Fermanian-Kammerer and Veronique Fischer.

Defect measures on graded lie groups.

Published in  Annali della Scuola Normale Superiore di Pisa. Preprint