TY - JOUR

T1 - Scaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets Around the Caustic

AU - Hanin, Boris

AU - Zelditch, Steve

AU - Zhou, Peng

N1 - Funding Information:
SZ is partially supported by NSF Grant DMS- 1541126, and BH is partially supported by NSF Grant DMS-1400822.
Publisher Copyright:
© 2016, Springer-Verlag Berlin Heidelberg.

PY - 2017/3/1

Y1 - 2017/3/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s00220-016-2807-4

DO - 10.1007/s00220-016-2807-4

M3 - Article

AN - SCOPUS:85000995834

VL - 350

SP - 1147

EP - 1183

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -