Planck-Scale Mass Equidistribution of Toral Laplace Eigenfunctions
We study the small scale distribution of the L 2 -mass of eigenfunctions of the Laplacian on the two-dimensional flat torus. Given an orthonormal basis of eigenfunctions, Lester and Rudnick (Commun. Math. Phys. 350(1):279–300, 2017 ) showed the existence of a density one subsequence whose L 2 -mass...
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| Veröffentlicht in: | Communications in mathematical physics 2017-10, Vol.355 (2), p.767-802 |
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| creator | Granville, Andrew Wigman, Igor |
| description | We study the small scale distribution of the
L
2
-mass of eigenfunctions of the Laplacian on the two-dimensional flat torus. Given an orthonormal basis of eigenfunctions, Lester and Rudnick (Commun. Math. Phys. 350(1):279–300,
2017
) showed the existence of a density one subsequence whose
L
2
-mass equidistributes more-or-less down to the Planck scale. We give a more precise version of their result showing equidistribution holds down to a small power of log above Planck scale, and also showing that the
L
2
-mass fails to equidistribute at a slightly smaller power of log above the Planck scale. This article rests on a number of results about the proximity of lattice points on circles, much of it based on foundational work of Javier Cilleruelo. |
| doi_str_mv | 10.1007/s00220-017-2953-3 |
| format | Article |
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L
2
-mass of eigenfunctions of the Laplacian on the two-dimensional flat torus. Given an orthonormal basis of eigenfunctions, Lester and Rudnick (Commun. Math. Phys. 350(1):279–300,
2017
) showed the existence of a density one subsequence whose
L
2
-mass equidistributes more-or-less down to the Planck scale. We give a more precise version of their result showing equidistribution holds down to a small power of log above Planck scale, and also showing that the
L
2
-mass fails to equidistribute at a slightly smaller power of log above the Planck scale. This article rests on a number of results about the proximity of lattice points on circles, much of it based on foundational work of Javier Cilleruelo.</description><identifier>ISSN: 0010-3616</identifier><identifier>EISSN: 1432-0916</identifier><identifier>DOI: 10.1007/s00220-017-2953-3</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Classical and Quantum Gravitation ; Complex Systems ; Eigenvectors ; Mathematical and Computational Physics ; Mathematical Physics ; Physics ; Physics and Astronomy ; Proximity ; Quantum Physics ; Relativity Theory ; Theoretical ; Toruses</subject><ispartof>Communications in mathematical physics, 2017-10, Vol.355 (2), p.767-802</ispartof><rights>The Author(s) 2017</rights><rights>Copyright Springer Science & Business Media 2017</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-e03b3bc96571d69647d1ccd3d3fbb20634ebe8ee9ab9c52ca049271eb98bfd123</citedby><cites>FETCH-LOGICAL-c316t-e03b3bc96571d69647d1ccd3d3fbb20634ebe8ee9ab9c52ca049271eb98bfd123</cites><orcidid>0000-0002-6152-4743</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00220-017-2953-3$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00220-017-2953-3$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Granville, Andrew</creatorcontrib><creatorcontrib>Wigman, Igor</creatorcontrib><title>Planck-Scale Mass Equidistribution of Toral Laplace Eigenfunctions</title><title>Communications in mathematical physics</title><addtitle>Commun. Math. Phys</addtitle><description>We study the small scale distribution of the
L
2
-mass of eigenfunctions of the Laplacian on the two-dimensional flat torus. Given an orthonormal basis of eigenfunctions, Lester and Rudnick (Commun. Math. Phys. 350(1):279–300,
2017
) showed the existence of a density one subsequence whose
L
2
-mass equidistributes more-or-less down to the Planck scale. We give a more precise version of their result showing equidistribution holds down to a small power of log above Planck scale, and also showing that the
L
2
-mass fails to equidistribute at a slightly smaller power of log above the Planck scale. This article rests on a number of results about the proximity of lattice points on circles, much of it based on foundational work of Javier Cilleruelo.</description><subject>Classical and Quantum Gravitation</subject><subject>Complex Systems</subject><subject>Eigenvectors</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematical Physics</subject><subject>Physics</subject><subject>Physics and Astronomy</subject><subject>Proximity</subject><subject>Quantum Physics</subject><subject>Relativity Theory</subject><subject>Theoretical</subject><subject>Toruses</subject><issn>0010-3616</issn><issn>1432-0916</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><recordid>eNp1kDFPwzAQhS0EEqXwA9giMRvu7NSJR6hKQSoCiTJbtnOpUkLS2snAvydRGFiYbnjfeyd9jF0j3CJAdhcBhAAOmHGhF5LLEzbDVAoOGtUpmwEgcKlQnbOLGPcAoIVSM_bwVtvGf_J3b2tKXmyMyerYV0UVu1C5vqvaJmnLZNsGWycbe6itp2RV7agp-8aPcbxkZ6WtI1393jn7eFxtl09887p-Xt5vuJeoOk4gnXReq0WGhdIqzQr0vpCFLJ0ToGRKjnIibZ32C-EtpFpkSE7nrixQyDm7mXYPoT32FDuzb_vQDC8NajHAOWoYKJwoH9oYA5XmEKovG74NghlVmUmVGVSZUZWRQ0dMnTiwzY7Cn-V_Sz_mtWub</recordid><startdate>20171001</startdate><enddate>20171001</enddate><creator>Granville, Andrew</creator><creator>Wigman, Igor</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-6152-4743</orcidid></search><sort><creationdate>20171001</creationdate><title>Planck-Scale Mass Equidistribution of Toral Laplace Eigenfunctions</title><author>Granville, Andrew ; Wigman, Igor</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-e03b3bc96571d69647d1ccd3d3fbb20634ebe8ee9ab9c52ca049271eb98bfd123</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>Classical and Quantum Gravitation</topic><topic>Complex Systems</topic><topic>Eigenvectors</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematical Physics</topic><topic>Physics</topic><topic>Physics and Astronomy</topic><topic>Proximity</topic><topic>Quantum Physics</topic><topic>Relativity Theory</topic><topic>Theoretical</topic><topic>Toruses</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Granville, Andrew</creatorcontrib><creatorcontrib>Wigman, Igor</creatorcontrib><collection>Springer Nature OA Free Journals</collection><collection>CrossRef</collection><jtitle>Communications in mathematical physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Granville, Andrew</au><au>Wigman, Igor</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Planck-Scale Mass Equidistribution of Toral Laplace Eigenfunctions</atitle><jtitle>Communications in mathematical physics</jtitle><stitle>Commun. Math. Phys</stitle><date>2017-10-01</date><risdate>2017</risdate><volume>355</volume><issue>2</issue><spage>767</spage><epage>802</epage><pages>767-802</pages><issn>0010-3616</issn><eissn>1432-0916</eissn><abstract>We study the small scale distribution of the
L
2
-mass of eigenfunctions of the Laplacian on the two-dimensional flat torus. Given an orthonormal basis of eigenfunctions, Lester and Rudnick (Commun. Math. Phys. 350(1):279–300,
2017
) showed the existence of a density one subsequence whose
L
2
-mass equidistributes more-or-less down to the Planck scale. We give a more precise version of their result showing equidistribution holds down to a small power of log above Planck scale, and also showing that the
L
2
-mass fails to equidistribute at a slightly smaller power of log above the Planck scale. This article rests on a number of results about the proximity of lattice points on circles, much of it based on foundational work of Javier Cilleruelo.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00220-017-2953-3</doi><tpages>36</tpages><orcidid>https://orcid.org/0000-0002-6152-4743</orcidid><oa>free_for_read</oa></addata></record> |
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| subjects | Classical and Quantum Gravitation Complex Systems Eigenvectors Mathematical and Computational Physics Mathematical Physics Physics Physics and Astronomy Proximity Quantum Physics Relativity Theory Theoretical Toruses |
| title | Planck-Scale Mass Equidistribution of Toral Laplace Eigenfunctions |
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