Counting the eigenvalues of elliptic operators is a classical subject with the remarkable Weyl asymptotics being a famous result. These problems are naturally stated in a semiclassical language. In this talk I will focus on a recent work in the 2-dimensional situation in the presence of a constant magnetic field and with Neumann boundary conditions. Suppose we count the eigenvalues up to a level $E$. In this case there is a transition when $E$ goes from being below to being above the lowest Landau level, that is the lowest eigenvalue for the problem on the entire plane. Below the lowest Landau level the asymptotics is of boundary type, i.e. $1$-dimensional in nature. Immediately above, it becomes $2$-dimensional. It has been an intriguing open problem to study the behavior exactly at the energy of the lowest Landau level and to determine which dimensionality it belongs to. I will report on new results on this question.
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This is joint work with M. Goffeng, A. Kachmar, and Mikael Persson-Sundqvist.