Scaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets Around the Caustic

Boris Hanin, Steve Zelditch, Peng Zhou

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Abstract

We study the scaling asymptotics of the eigenspace projection kernels Π ħ , E(x, y) of the isotropic Harmonic Oscillator H^ ħ= - ħ2Δ + | x| 2 of eigenvalue E=ħ(N+d2) in the semi-classical limit ħ→ 0 . The principal result is an explicit formula for the scaling asymptotics of Π ħ , E(x, y) for x, y in a ħ2 / 3 neighborhood of the caustic CE as ħ→ 0. The scaling asymptotics are applied to the distribution of nodal sets of Gaussian random eigenfunctions around the caustic as ħ→ 0 . In previous work we proved that the density of zeros of Gaussian random eigenfunctions of H^ ħ have different orders in the Planck constant ħ in the allowed and forbidden regions: In the allowed region the density is of order ħ- 1 while it is ħ- 1 / 2 in the forbidden region. Our main result on nodal sets is that the density of zeros is of order ħ-23 in an ħ23 -tube around the caustic. This tube radius is the ‘critical radius’. For annuli of larger inner and outer radii ħα with 0<α<23 we obtain density results that interpolate between this critical radius result and our prior ones in the allowed and forbidden region. We also show that the Hausdorff (d−2)-dimensional measure of the intersection of the nodal set with the caustic is of order ħ-23.

Original languageEnglish (US)
Pages (from-to)1147-1183
Number of pages37
JournalCommunications In Mathematical Physics
Volume350
Issue number3
DOIs
StatePublished - Mar 1 2017
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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