Microlocal analysis and its applications in quantum resonances, partial differential equations with analytic or Gevrey regularity, and several complex variables.

About the Project

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.

Funding Notes

Funding is available through the School of Mathematics for a suitably strong candidate. The scholarship will cover tuition fees, training support, and a stipend at standard rates for 3-3.5 years.

Candidates are encouraged to make an informal inquiry with Dr Haoren Xiong.

References

Xiong, H., Pseudodifferential operators with formal Gevrey symbols and symbolic calculus, arXiv: 2603.07693.
Xiong, H., Boundary spectral estimates for semiclassical Gevrey operators. J. Spectr. Theory 15 (2025), no. 4, pp. 1503–1522.
Xiong, H. and Xu, H., Semiclassical asymptotics for Bergman projections with Gevrey weights, arXiv: 2403.14157.
Xiong, H., Generic simplicity of resonances in obstacle scattering, Trans. Amer. Math. Soc. 376 (2023), 4301-4319.