At a Glance
- Tasks: Explore microlocal analysis and its applications in quantum resonances and PDEs.
- Company: Join a leading academic institution with a focus on mathematical physics.
- Benefits: Fully funded PhD scholarship covering tuition, training support, and stipend for 3-3.5 years.
- Other info: Work under the guidance of Dr Haoren Xiong in a dynamic research environment.
- Why this job: Dive into cutting-edge research and make significant contributions to quantum mechanics and geometry.
- Qualifications: Strong background in mathematics and interest in partial differential equations.
The predicted salary is between 18000 - 25000 £ per year.
Microlocal analysis explores the classical/quantum correspondence in partial differential equations (PDEs).
For example, the concept of symbol quantisation in the theory of pseudodifferential operators links classical observables (functions on phase space) with quantum operators.
Microlocal analysis also provides essential tools to rigorously define, compute, and analyse resonances, which describe the oscillation and decay of waves in systems where energy can escape to infinity, replacing eigenvalues and eigenfunction expansions in non-compact domains.
The consideration of analytic or Gevrey regularity in the study of PDEs has attracted considerable attention in recent years.
The aim of this project is to develop microlocal analysis techniques in analytic and Gevrey settings.
The analytic scheme often yields exponentially accurate approximation or controls in PDEs.
Gevrey regularity allows one to go beyond the smooth theory and obtain quantitative remainder estimates, while retaining useful tools such as cutoffs and partitions of unity that are unavailable in the analytic setting.
We expect to apply these techniques to geometry, quantum mechanics, and other fields in mathematical physics.
The project will be supervised by Dr Haoren Xiong (h. xiong@bham. ac. uk).
Funding notes
Funding is available through the School of Mathematics for a suitably strong candidate.
The scholarship will cover tuition fees, training support, anda stipend at standard rates for 3-3.5 years.
- References
- Xiong, H., Pseudodifferential operatorswithformal Gevreysymbolsandsymbolic calculus, ar
- Xiong, H., Boundary spectral estimates for semiclassical Gevreyoperators. J. Spectr. Theory 15 (2025), no. 4, pp. 1503–1522.
- Xiong, H. and Xu, H., Semiclassical asymptotics for Bergman projectionswith Gevreyweights, ar
- Xiong, H., Generic simplicity ofresonancesinobstacle scattering, Trans. Amer. Math. Soc. 376 (2023), 4301-4319.
- #J-18808-Ljbffr
PhD Studentship: Microlocal analysis and its applications in quantum resonances, partial differ[...] in Birmingham employer: University of Birmingham
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