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Time-band limiting: searching for a miracle, armed with a new motivation

Banff International Research Station for Mathematical Innovation and Discovery

The subject starts with C. Shannon questions at Bell Labs and the amazing answers found in the 60's by Slepian, Landau and Pollak. The crux of the matter is that in some naturally appearing problems in signal processing a certain integral operator admits an explicit commuting second order differential one, or a full matrix admits a commuting tridiagonal one. These operators are parametrized by the duration of the signal and its bandwidth. I have been trying to understand and extend this miracle for a long time, mainly in "non translation invariant cases". All the examples from Bell Labs involve Fourier analysis in different setups. I have managed to extend this to a few other cases. My "new motivation" comes from looking at two papers: "Maximal violations of Bell inequalities by position measurements" , J. Kiukas and R. Werner 2009 and "Properties of the entanglement Hamiltonian for finite free-fermion chains", V. Eisler and I. Peschel 2018. Both of these papers exploit the commutativity property in some physically important cases. My hope is that some members of the audience may point out a few more cases where this very exceptional mathematical phenomenon plays a physically relevant role.

Mathematics; Quantum theory; Mathematical physics

video/mp4

Author affiliation: University of California, Berkeley

2019-10-24

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/72042

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/