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# Boris Hanin (MIT), Analysis Seminar

## November 9, 2016 @ 4:00 pm - 5:00 pm

**Title: **Airy Scaling for Spectral Projectors of the Harmonic Oscillator

**Abstract: **Energy E eigenfunctions of any Schrodinger operator are rapidly oscillating in the classically allowed region {V(x)<E} and exponentially decaying in the classically forbidden region {V(x)>E}. When d=1, eigenspaces are simple and the Airy-type transition of individual eigenfunctions near the caustic {V(x)=E} is classical. I will present joint work with S. Zelditch and P. Zhou, which extends this result to higher dimensions in the model case of the harmonic oscillator (V(x)=|x|^2). Rather than work with individual eigenfunctions, we obtain an explicit formula for the scaling limit around the caustic of the fixed energy spectral projector. I will explain how this formula allows one to study the nodal sets of random Hermite functions both near the caustic and deep into the forbidden region.