The Øresund Seminar on Analysis, PDE & Related Topics

Abstract: 

Magnetic Schrödinger operators in the unit disk provide a useful testing ground for spectral questions. The symmetry of the disk allows for a much more detailed analysis than is possible for typical domains, but it also creates special difficulties and phenomena that do not appear in generic situations.

The talk will take as its starting point an image from a 1965 work of Saint-James, an image that suggests several natural questions about the spectrum. I will describe some known spectral properties of these operators, explain what numerical computations seem to suggest, and discuss several questions that remain open.

Abstract: This talk is going to be about curve shortening flow in high codimension for arcs with free boundary meeting a fixed smooth barrier orthogonally. We prove dilation-invariant curvature and higher-derivative estimates up to the boundary using a Stahl-type localised maximum principle and an adapted cut-off. Using a reflected Gaussian entropy and blow-up analysis, Type I boundary singularities yield a shrinking semicircle model after reflection. Type II blow-ups give a Grim Reaper translator, which is ruled out under a free-boundary entropy bound $<2$. Hence in the low-entropy regime the flow either converges to the orthogonal chord or has only semicircle boundary singularities. This is a joint work with Huy T. Nguyen.

Abstract: 

Given a compact set K ⊆ C containing infinitely many points, there is a unique monic polynomial of degree n that minimizes the uniform norm on K. This polynomial is called the Chebyshev polynomial of degree n for K. A classical problem is to understand how the geometric properties of K are reflected in the asymptotic behavior of the associated Chebyshev polynomials as n grows.

This picture is more or less complete when K consists of smooth Jordan curves that are mutually disjoint. Faber considered the case of a single analytic Jordan curve in 1920, and Widom considered several curves in 1969.

Chebyshev polynomials for Jordan arcs have proven more elusive. Widom conjectured in 1969 that the interval should provide the model behavior in this setting. However, this conjecture was shown to be false by a counterexample involving arcs on the unit circle. Instead, the two sides of the arc turn out to play a crucial role. In this talk, I will explain recent progress toward a revised conjecture by Christiansen, Simon, and Zinchenko.

This is based on joint work with Benedikt Buchecker, Benjamin Eichinger and Aron Wennman.

Abstract: 

For a nice function g we consider the set of its time-frequency shifts g_{n,m}(x) = g(x-an) exp(2πi bmx) over integers n, m and ask if it forms a frame for L^2(R), that is if for all functions f in L^2(R) the sum of |<f, g_{n,m}>|^2 is proportional to ||f||^2.

Density theorem says that if ab > 1 then this is never so. If the function g is supported on [c, d] then we also have a necessary condition a < d-c as otherwise the supports of the functions g_{n,m} do not even cover the whole R. In this talk we will show that if the function g is generic then these two conditions are also sufficient as long as the density ab is irrational.

The talk is based on a joint work with Yurii Belov.