Project description
Semiclassical analysis studies high-frequency limits in which quantum dynamics are related to classical Hamiltonian dynamics. For elliptic operators on Riemannian manifolds, this theory is well developed. This project extended semiclassical and microlocal methods to subelliptic settings, where the relevant non-elliptic operators reflect underlying structures such as sub-Riemannian and contact geometries.
Using representation theory and pseudodifferential calculi on graded Lie groups, nilmanifolds and filtered manifolds, the project investigated quantum and semiclassical limits, spectral asymptotics and Weyl laws, and developed pseudodifferential calculi adapted to the underlying geometry.
Project team and collaborators
Véronique Fischer, Principal Investigator
Clotilde Fermanian Kammerer, Co-Investigator
Steven Flynn, Postdoctoral Research Associate on the project, 22 January 2021 to 15 September 2023
Søren Mikkelsen, Postdoctoral Research Associate on the project, 14 February 2023 to 31 August 2024
Collaborators: Lino Benedetto · Cyril Letrouit · Francesca Tripaldi
Research outputs
C. Fermanian Kammerer, V. Fischer and S. Flynn, Quantization on Filtered Manifolds. Forthcoming, Memoirs of the European Mathematical Society. arXiv:2412.17448
V. Fischer and S. Mikkelsen, Semiclassical Functional Calculus on Nilpotent Lie Groups and their Compact Nilmanifolds. Analysis and Mathematical Physics 15 (2025), no. 3, Paper No. 60. arXiv:2409.05520
S. Mikkelsen, Sharp Semiclassical Spectral Asymptotics for Schrödinger Operators with Non-smooth Potentials. Journal of Spectral Theory 15 (2025), no. 2, 819–846. arXiv:2309.12015
S. Mikkelsen, Sharp Semiclassical Spectral Asymptotics for Local Magnetic Schrödinger Operators on R^d Without Full Regularity. Annales Henri Poincaré 26 (2025), 1865–1906. arXiv:2309.03716
L. Benedetto, C. Fermanian Kammerer and V. Fischer, Wick Quantization on Groups and Application to Gårding Inequalities. Journal of the Mathematical Society of Japan 77 (2025), no. 3, 797–832. arXiv:2307.15352
S. Flynn, Singular Value Decomposition for the X-Ray Transforms on the Reduced Heisenberg Group, and a Two-Radius Theorem. In Extended Abstracts 2021/2022 (GMC 2021), Trends in Mathematics, vol. 3, 15–26, Birkhäuser, Cham, 2024. arXiv:2305.04126
V. Fischer and F. Tripaldi, Subcomplexes on Filtered Riemannian Manifolds. Preprint. arXiv:2312.02342
S. Flynn, The Sub-Riemannian X-ray Transform on H-type Groups: Fourier-Slice Theorems and Injectivity Sets. Preprint. arXiv:2312.00594
C. Fermanian Kammerer, V. Fischer and S. Flynn, Some Remarks on Semi-classical Analysis on Two-Step Nilmanifolds. In Quantum Mathematics I, Springer INdAM Series 57, 129–162, Springer, Singapore, 2023. arXiv:2211.14273
V. Fischer and F. Tripaldi, An Alternative Construction of the Rumin Complex on Homogeneous Nilpotent Lie Groups. Advances in Mathematics 429 (2023), Paper No. 109192. arXiv:2208.11118
C. Fermanian Kammerer, V. Fischer and S. Flynn, Geometric Invariance of the Semiclassical Calculus on Nilpotent Graded Lie Groups. Journal of Geometric Analysis 33 (2023), no. 4, Paper No. 127. arXiv:2112.11509
V. Fischer, Asymptotics and Zeta Functions on Compact Nilmanifolds. Journal de Mathématiques Pures et Appliqués 160 (2022), 1–28. arXiv:2112.01246
V. Fischer, Towards Semi-classical Analysis for Sub-elliptic Operators. Bruno Pini Mathematical Analysis Seminar 12 (2022), 31–52. arXiv:2111.09854
V. Fischer, Semiclassical Analysis on Compact Nil-manifolds. Preprint. arXiv:2101.07027
C. Fermanian Kammerer and C. Letrouit, Observability and Controllability for the Schrödinger Equation on Quotients of Groups of Heisenberg Type. Journal de l'École polytechnique — Mathématiques 8 (2021), 1459–1513. arXiv:2009.13877
C. Fermanian Kammerer and V. Fischer, Quantum Evolution and Sub-Laplacian Operators on Groups of Heisenberg Type. Journal of Spectral Theory 11 (2021), 1313–1367. arXiv:1910.14486
Lecture notes and dissemination
V. Fischer, Harmonic Analysis on the Heisenberg Group and Related Topics, lecture notes for the Summer School on Singular Integrals on Nilpotent Lie Groups and Related Topics, Göttingen, 19–23 September 2022. Lecture notes (PDF)
C. Fermanian Kammerer, Semi-classical Analysis on Graded Lie Groups, lecture notes for the Spring School on Modern Aspects of Analysis on Lie Groups, Göttingen, April 2024. Lecture notes (PDF)
V. Fischer, Quantum Limits for Sub-Elliptic Operators, Oberwolfach Report 24/2023, workshop Hypoelliptic Operators in Geometry, 21–26 May 2023. Oberwolfach report
Funding acknowledgement
This project was supported by the Leverhulme Trust through Research Project Grant RPG-2020-037 (£318,871). Further details are available on the University of Bath Research Portal.