Scaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets Around the Caustic
Author(s)Hanin, Boris; Zelditch, Steve; Zhou, Peng
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We study the scaling asymptotics of the eigenspace projection kernels Π[subscript ℏ,E](x,y) of the isotropic Harmonic Oscillator [^ over H][subscript ℏ]=−ℏ[superscript 2]Δ+|x|[superscript 2] of eigenvalue E=ℏ(N+d/2) in the semi-classical limit ℏ→0. The principal result is an explicit formula for the scaling asymptotics of Π[subscript ℏ,E](x,y) for x, y in a ℏ[superscript 2/3] neighborhood of the caustic C[subscript E] 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 [^ over H][subscript 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[superscript −1] while it is ℏ[superscript −1/2] in the forbidden region. Our main result on nodal sets is that the density of zeros is of order ℏ[superscript −2/3] in an ℏ[superscript 2/3] -tube around the caustic. This tube radius is the ‘critical radius’. For annuli of larger inner and outer radii ℏ[superscript α] with 0<α<2/3 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[superscript −2/3] .
Communications in Mathematical Physics
Springer Berlin Heidelberg
Hanin, Boris, Steve Zelditch, and Peng Zhou. “Scaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets Around the Caustic.” Communications in Mathematical Physics 350.3 (2017): 1147–1183.
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