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Quantum limits for sub-elliptic operators

Department of Mathematical Sciences

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Dr Veronique Fischer No more applications being accepted Funded PhD Project (European/UK Students Only)

About the Project

The University of Bath is inviting applications for a funded PhD project in the Department of Mathematical Sciences


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.


Applicants will have obtained, or expect to receive shortly, a Master degree in Mathematics with a first or a 2.1, or an equivalent qualification. They will have a keen interest in various aspects of mathematical analysis and be enthusiastic about the project. Previous relevant experiences, for instance in spectral, semi-classical or/and harmonic analysis or/and in Lie groups, will be an advantage.

Non-UK applicants must meet our English language entry requirement


Information enquiries are welcomed and should be addressed to Dr Veronique Fischer ([Email Address Removed]).

Formal applications should be made via the University of Bath’s online application form for a PhD in Mathematics:

Please ensure that you quote the supervisor’s name and project title in the ‘Your research interests’ section.

More information about applying for a PhD at Bath may be found here:

NOTE: Applications will be reviewed on an ongoing basis until the position is filled; therefore, it is possible that applications may close before the advertised deadline and early application is strongly recommended.

Anticipated start date: By mutual agreement. Earliest possible start date is 28 September 2020.

Funding Notes

UK and EU citizens applying for this project will be considered for a studentship covering UK/EU tuition fees and providing a tax-free maintenance stipend (£15,285 per annum, 2020/21 rate) for a period of up to 3.5 years.

The studentship is financed by a Leverhulme research project grant which provides additional funds for training and travel.

Unfortunately, applicants who are classed as 'Overseas' for fee paying purposes are not eligible for funding and will not be considered unless they can provide documentary evidence of their ability to fully self-fund their studies (Overseas tuition fees, research expenses/bench fees and living costs).


• Clotilde Fermanian-Kammerer and Veronique Fischer. Quantum evolution and sub-Laplacian operators on groups of Heisenberg type. To appear in Journal of Spectral Theory (37 pages). Arxiv 1910.14486

• Clotilde Fermanian-Kammerer and Veronique Fischer. Defect measures on graded Lie groups. To appear in Annali della Scuola Normale Superiore di Pisa (85 pages). ArXiv 1707.04002

• Clotilde Fermanian-Kammerer and Veronique Fischer. Semi-classical analysis on H-type groups. Science China Mathematics (2019) 62: 1057.

• Veronique Fischer and Michael Ruzhansky. Quantization on nilpotent Lie groups (557 pages). Progress in Mathematics 314, Birkhäuser Springer, 2016. Winner of the Ferran Sunyer I Balaguer prize in 2014
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