2,371 research outputs found
Nodal Sets of Random Eigenfunctions for the Isotropic Harmonic Oscillator
We consider Gaussian random eigenfunctions (Hermite functions) of fixed
energy level of the isotropic semi-classical Harmonic Oscillator on . We calculate the expected density of zeros of a random eigenfunction in
the semi-classical limit In the allowed region the density is of
order while in the forbidden region the density is of order
. The computer graphics due to E.J. Heller illustrate this
difference in "frequency" between the allowed and forbidden nodal sets.Comment: 3 figures, 2 due to E. J. Heller. Corrected the calculation in the
forbidden regio
Scaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets Around the Caustic
We study the scaling asymptotics of the eigenspace projection kernels
of the isotropic Harmonic Oscillator of eigenvalue in the semi-classical limit
. The principal result is an explicit formula for the scaling
asymptotics of for in a neighborhood
of the caustic as The scaling asymptotics are
applied to the distribution of nodal sets of Gaussian random eigenfunctions
around the caustic as . In previous work we proved that the
density of zeros of Gaussian random eigenfunctions of have
different orders in the Planck constant in the allowed and forbidden
regions: In the allowed region the density is of order while it is
in the forbidden region. Our main result on nodal sets is that
the density of zeros is of order in an
-tube around the caustic. This tube radius is the
`critical radius'. For annuli of larger inner and outer radii
with we obtain density results which interpolate
between this critical radius result and our prior ones in the allowed and
forbidden region. We also show that the Hausdorff -dimensional measure
of the intersection of the nodal set with the caustic is of order .Comment: v3. Accepted to Communications in Mathematical Physic
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