Research

I am broadly interested in noncommutative geometry and topological aspects of mathematical physics, with my research focusing on K-theoretic methods of obtaining topological invariants from scattering systems. In particular, using spectral and scattering information to recognise index theoretic data in quantum systems. I am also interested in applying techniques from noncommutative geometry to obtain explicit and computable formulae for this index data. 

So far, my research has focused on recognising Levinson’s theorem in Euclidean scattering as an index pairing in order to build a local index formula for computing the number of bound states of a quantum mechanical system, as well as investigating the contributions of anomolous behaviour such as zero-energy resonances to such formulae. I have also worked with spectral flow and the spectral shift function for Schrödinger operators.

All of my papers can be found (in preprint form) on arXiv here (see also my ORCID profile and my Google Scholar profile). 

[1] A. Alexander, A. Rennie. `Levinson's theorem as an index pairing', J. Funct. Anal., 286 (5), 2024 (available here).

[2] A. Alexander, D. T. Nguyen, S. Richard, A. Rennie. `Levinson's theorem for two-dimensional scattering systems: it was a surprise, it is now topological!', J. Spect. Theor., 14 (3), 2024, 991-1031 (available here).

[3] A. Alexander. `Trace formula and Levinson's theorem in the presence of resonances', Rev. Math. Phys., DOI: 10.1142/S0129055X24500363, 2024 (preprint available here).

[4] A. Alexander, A. Rennie. `The structure of the wave operator in four dimensions in the presence of resonances', Lett. Math. Phys., 114 (122), 2024 (available here).

[1] A. Alexander. `Topological Levinson's theorem via index pairings and spectral flow', PhD thesis, University of Wollongong, 2024. (available here).

[2] A. Alexander, A. Rennie. `Spectral flow and Levinson's theorem for Schrödinger operators', arXiv preprint: https://arxiv.org/abs/2405.19571, 2024.