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dc.contributor.authorHanin, Boris
dc.contributor.authorZelditch, Steve
dc.contributor.authorZhou, Peng
dc.date.accessioned2017-03-22T18:43:00Z
dc.date.available2017-10-01T05:00:06Z
dc.date.issued2016-12
dc.date.submitted2016-02
dc.identifier.issn0010-3616
dc.identifier.issn1432-0916
dc.identifier.urihttp://hdl.handle.net/1721.1/107651
dc.description.abstractWe 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.sponsorshipNational Science Foundation (U.S.) (Grant DMS-1400822)en_US
dc.publisherSpringer Berlin Heidelbergen_US
dc.relation.isversionofhttp://dx.doi.org/10.1007/s00220-016-2807-4en_US
dc.rightsArticle 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.sourceSpringer Berlin Heidelbergen_US
dc.titleScaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets Around the Causticen_US
dc.typeArticleen_US
dc.identifier.citationHanin, 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.departmentMassachusetts Institute of Technology. Department of Mathematics
dc.contributor.mitauthorHanin, Boris
dc.relation.journalCommunications in Mathematical Physicsen_US
dc.eprint.versionAuthor's final manuscripten_US
dc.type.urihttp://purl.org/eprint/type/JournalArticleen_US
eprint.statushttp://purl.org/eprint/status/PeerRevieweden_US
dc.date.updated2017-02-11T04:58:21Z
dc.language.rfc3066en
dc.rights.holderSpringer-Verlag Berlin Heidelberg
dspace.orderedauthorsHanin, Boris; Zelditch, Steve; Zhou, Pengen_US
dspace.embargo.termsNen
dc.identifier.orcidhttps://orcid.org/0000-0001-5911-1432
mit.licensePUBLISHER_POLICYen_US
mit.metadata.statusComplete


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