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Complex Cauchy matrices, inverse spectral problems and colourful invariant tori

Patrick Gerard
Universitè Paris-Sud
Tuesday, January 29, 2019 - 16:30
The Cauchy matrix $c_{jk}=(a_j+b_k)^{-1}$ is well known in linear algebra, with an explicit formula for its inverse. A less standard fact is he that the complex Cauchy matrix given by
is invertible under simple generic assumptions on the complex numbers $a_j$, $b_k$. I will start with an elementary proof of this fact, and then explain the role played by these matrices in inverse spectral problems for some special operators on the Hilbert space $\ell ^2$, called Hankel operators. The solutions of such inverse spectral problems are infinite dimensional tori in a functional space on the circle, the Hardy space.
These tori are supporting the dynamics of some integrable infinite dimensional system which enjoys surprising regularity properties.
This talk is inspired from a series of jointworks with Alexander Pushnitski (King's College, London) and Sandrine Grellier (Orléans).

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