Non-self-adjoint spectral problems

Non-self-adjoint spectral theory is not yet a coherent subject, in the sense that there is no analogue of the spectral theorem that can act as a basis for further research. The very high instability of eigenvalues under small perturbations often affects the analysis of particular models. The number of these understood is expanding rapidly and will continue to do so into the foreseeable future.

We describe joint work with Michael Levitin on the N → ∞ asymptotic spectral behaviour of a particular family of large non-self-adjoint matrices Ac,N associated with a self-adjoint linear pencil. Crucial insights were obtained by numerical experiments, even though the final analysis does not use rely on numerics. The problem is a matrix analogue of an indefinite self-adjoint linear pencil that concerns a Dirac operator with an indefinite potential. In some sense it is the simplest matrix example of its type, but its behaviour is still far more complex than one might expect. The eigenvalues of the matrix Ac,N converge to the real axis as N → ∞, but the details of the convergence depend strongly on the choice of the real parameter c, in a way that presently defies understanding, even at a numerical level.

About the speaker

Professor Brian Davies has a post-retirement, part time position at King's College, London. He was the President of the London Mathematical Society from 2007 to 2009; honours include a Fellowship of the Royal Society since 1995.

He is the author of more than 200 mathematical papers, mostly devoted to analysis, heat equations and non-self-adjoint spectral theory, and he has written 7 books, including 'Linear Operators and their Spectra'; his interests encompass the philosophy of mathematics - as seen in 'Science in the Looking Glass' and 'Why Beliefs Matter'. He is also well known in the quantum physics community for work that he did in the 1970s on quantum measurement and open quantum systems.

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