dc.contributor.author | Hanin, Boris | |
dc.contributor.author | Zelditch, Steve | |
dc.contributor.author | Zhou, Peng | |
dc.date.accessioned | 2017-03-22T18:43:00Z | |
dc.date.available | 2017-10-01T05:00:06Z | |
dc.date.issued | 2016-12 | |
dc.date.submitted | 2016-02 | |
dc.identifier.issn | 0010-3616 | |
dc.identifier.issn | 1432-0916 | |
dc.identifier.uri | http://hdl.handle.net/1721.1/107651 | |
dc.description.abstract | 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] . | en_US |
dc.description.sponsorship | National Science Foundation (U.S.) (Grant DMS-1400822) | en_US |
dc.publisher | Springer Berlin Heidelberg | en_US |
dc.relation.isversionof | http://dx.doi.org/10.1007/s00220-016-2807-4 | en_US |
dc.rights | Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. | en_US |
dc.source | Springer Berlin Heidelberg | en_US |
dc.title | Scaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets Around the Caustic | en_US |
dc.type | Article | en_US |
dc.identifier.citation | 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. | en_US |
dc.contributor.department | Massachusetts Institute of Technology. Department of Mathematics | |
dc.contributor.mitauthor | Hanin, Boris | |
dc.relation.journal | Communications in Mathematical Physics | en_US |
dc.eprint.version | Author's final manuscript | en_US |
dc.type.uri | http://purl.org/eprint/type/JournalArticle | en_US |
eprint.status | http://purl.org/eprint/status/PeerReviewed | en_US |
dc.date.updated | 2017-02-11T04:58:21Z | |
dc.language.rfc3066 | en | |
dc.rights.holder | Springer-Verlag Berlin Heidelberg | |
dspace.orderedauthors | Hanin, Boris; Zelditch, Steve; Zhou, Peng | en_US |
dspace.embargo.terms | N | en |
dc.identifier.orcid | https://orcid.org/0000-0001-5911-1432 | |
mit.license | PUBLISHER_POLICY | en_US |
mit.metadata.status | Complete | |