Håkan Hedenmalm (KTH, Stockholm) Leverhulme lecture

Soft Riemann-Hilbert problems and planar orthogonal polynomials


It is known that planar orthogonal polynomials with respect to exponentially varying weights can be characterized in terms of a matrix dbar-problem. In recent work with A. Wennman, an asymptotic expansion for the orthogonal polynomials was found. The proof required scaffolding in terms of the construction of an invariant flow. Here, we find a direct approach in terms of a guess for the Cauchy potential in the Riemann-Hilbert problem. This seems to connect with integrable hierarchies but that aspect remains to explore. The solution allows for better control of the error term, and the main underlying equation is solved by finite Neumann series expansion.