Spectral Simplicity and Asymptotic Separation of Variables
We describe a method for comparing the spectra of two real-analytic families, ( a t ) and ( q t ), of quadratic forms that both degenerate as a positive parameter t tends to zero. We suppose that the family ( a t ) is amenable to ‘separation of variables’ and that each eigenspace of a t is 1-dimensi...
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| Veröffentlicht in: | Communications in mathematical physics 2011, Vol.302 (2), p.291-344 |
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| container_title | Communications in mathematical physics |
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| creator | Hillairet, Luc Judge, Chris |
| description | We describe a method for comparing the spectra of two real-analytic families, (
a
t
) and (
q
t
), of quadratic forms that both degenerate as a positive parameter
t
tends to zero. We suppose that the family (
a
t
) is amenable to ‘separation of variables’ and that each eigenspace of
a
t
is 1-dimensional for some
t
. We show that if (
q
t
) is asymptotic to (
a
t
) at first order as
t
→ 0, then the eigenspaces of (
q
t
) are also 1-dimensional for all but countably many
t
. As an application, we prove that for the generic triangle (simplex) in Euclidean space (constant curvature space form) each eigenspace of the Laplacian acting on Dirichlet functions is 1-dimensional. |
| doi_str_mv | 10.1007/s00220-010-1185-6 |
| format | Article |
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a
t
) and (
q
t
), of quadratic forms that both degenerate as a positive parameter
t
tends to zero. We suppose that the family (
a
t
) is amenable to ‘separation of variables’ and that each eigenspace of
a
t
is 1-dimensional for some
t
. We show that if (
q
t
) is asymptotic to (
a
t
) at first order as
t
→ 0, then the eigenspaces of (
q
t
) are also 1-dimensional for all but countably many
t
. As an application, we prove that for the generic triangle (simplex) in Euclidean space (constant curvature space form) each eigenspace of the Laplacian acting on Dirichlet functions is 1-dimensional.</description><identifier>ISSN: 0010-3616</identifier><identifier>EISSN: 1432-0916</identifier><identifier>DOI: 10.1007/s00220-010-1185-6</identifier><identifier>CODEN: CMPHAY</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer-Verlag</publisher><subject>Algebra ; Analysis of PDEs ; Classical and Quantum Gravitation ; Complex Systems ; Differential geometry ; Exact sciences and technology ; Geometry ; Mathematical and Computational Physics ; Mathematical methods in physics ; Mathematical Physics ; Mathematics ; Number theory ; Other topics in mathematical methods in physics ; Physics ; Physics and Astronomy ; Quantum Physics ; Relativity Theory ; Sciences and techniques of general use ; Spectral Theory ; Theoretical</subject><ispartof>Communications in mathematical physics, 2011, Vol.302 (2), p.291-344</ispartof><rights>Springer-Verlag 2011</rights><rights>2015 INIST-CNRS</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c352t-ed248b8474b0fcf3ae81c9f18cb5880b778c1f5fddf3044aa038f8cf15f58c663</citedby><cites>FETCH-LOGICAL-c352t-ed248b8474b0fcf3ae81c9f18cb5880b778c1f5fddf3044aa038f8cf15f58c663</cites><orcidid>0009-0009-6010-151X</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/s00220-010-1185-6$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00220-010-1185-6$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>230,314,776,780,881,27901,27902,41464,42533,51294</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=23864378$$DView record in Pascal Francis$$Hfree_for_read</backlink><backlink>$$Uhttps://hal.science/hal-00445680$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Hillairet, Luc</creatorcontrib><creatorcontrib>Judge, Chris</creatorcontrib><title>Spectral Simplicity and Asymptotic Separation of Variables</title><title>Communications in mathematical physics</title><addtitle>Commun. Math. Phys</addtitle><description>We describe a method for comparing the spectra of two real-analytic families, (
a
t
) and (
q
t
), of quadratic forms that both degenerate as a positive parameter
t
tends to zero. We suppose that the family (
a
t
) is amenable to ‘separation of variables’ and that each eigenspace of
a
t
is 1-dimensional for some
t
. We show that if (
q
t
) is asymptotic to (
a
t
) at first order as
t
→ 0, then the eigenspaces of (
q
t
) are also 1-dimensional for all but countably many
t
. As an application, we prove that for the generic triangle (simplex) in Euclidean space (constant curvature space form) each eigenspace of the Laplacian acting on Dirichlet functions is 1-dimensional.</description><subject>Algebra</subject><subject>Analysis of PDEs</subject><subject>Classical and Quantum Gravitation</subject><subject>Complex Systems</subject><subject>Differential geometry</subject><subject>Exact sciences and technology</subject><subject>Geometry</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematical methods in physics</subject><subject>Mathematical Physics</subject><subject>Mathematics</subject><subject>Number theory</subject><subject>Other topics in mathematical methods in physics</subject><subject>Physics</subject><subject>Physics and Astronomy</subject><subject>Quantum Physics</subject><subject>Relativity Theory</subject><subject>Sciences and techniques of general use</subject><subject>Spectral Theory</subject><subject>Theoretical</subject><issn>0010-3616</issn><issn>1432-0916</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2011</creationdate><recordtype>article</recordtype><recordid>eNp9kDFPwzAQhS0EEqXwA9iyMDAE7uzEcdmqCihSJYYCq3VxbHCVJpEdkPrvSRTUkel09773pHuMXSPcIUBxHwE4hxQQUkSVp_KEzTATPIUFylM2g1EREuU5u4hxBwALLuWMPWw7a_pAdbL1-672xveHhJoqWcbDvuvb3ptkazsK1Pu2SVqXfFDwVNY2XrIzR3W0V39zzt6fHt9W63Tz-vyyWm5SI3Lep7bimSpVVmQlOOMEWYVm4VCZMlcKyqJQBl3uqsoJyDIiEMop4zB3uTJSijm7nXK_qNZd8HsKB92S1-vlRo83GGy5VPCDA4sTa0IbY7DuaEDQY1F6KkrDuA9F6TH_ZvJ0FA3VLlBjfDwauVAyE4UaOD5xcZCaTxv0rv0OzfD6P-G_ipZ3TQ</recordid><startdate>2011</startdate><enddate>2011</enddate><creator>Hillairet, Luc</creator><creator>Judge, Chris</creator><general>Springer-Verlag</general><general>Springer</general><general>Springer Verlag</general><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>1XC</scope><orcidid>https://orcid.org/0009-0009-6010-151X</orcidid></search><sort><creationdate>2011</creationdate><title>Spectral Simplicity and Asymptotic Separation of Variables</title><author>Hillairet, Luc ; Judge, Chris</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c352t-ed248b8474b0fcf3ae81c9f18cb5880b778c1f5fddf3044aa038f8cf15f58c663</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2011</creationdate><topic>Algebra</topic><topic>Analysis of PDEs</topic><topic>Classical and Quantum Gravitation</topic><topic>Complex Systems</topic><topic>Differential geometry</topic><topic>Exact sciences and technology</topic><topic>Geometry</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematical methods in physics</topic><topic>Mathematical Physics</topic><topic>Mathematics</topic><topic>Number theory</topic><topic>Other topics in mathematical methods in physics</topic><topic>Physics</topic><topic>Physics and Astronomy</topic><topic>Quantum Physics</topic><topic>Relativity Theory</topic><topic>Sciences and techniques of general use</topic><topic>Spectral Theory</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Hillairet, Luc</creatorcontrib><creatorcontrib>Judge, Chris</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Hyper Article en Ligne (HAL)</collection><jtitle>Communications in mathematical physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Hillairet, Luc</au><au>Judge, Chris</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Spectral Simplicity and Asymptotic Separation of Variables</atitle><jtitle>Communications in mathematical physics</jtitle><stitle>Commun. Math. Phys</stitle><date>2011</date><risdate>2011</risdate><volume>302</volume><issue>2</issue><spage>291</spage><epage>344</epage><pages>291-344</pages><issn>0010-3616</issn><eissn>1432-0916</eissn><coden>CMPHAY</coden><abstract>We describe a method for comparing the spectra of two real-analytic families, (
a
t
) and (
q
t
), of quadratic forms that both degenerate as a positive parameter
t
tends to zero. We suppose that the family (
a
t
) is amenable to ‘separation of variables’ and that each eigenspace of
a
t
is 1-dimensional for some
t
. We show that if (
q
t
) is asymptotic to (
a
t
) at first order as
t
→ 0, then the eigenspaces of (
q
t
) are also 1-dimensional for all but countably many
t
. As an application, we prove that for the generic triangle (simplex) in Euclidean space (constant curvature space form) each eigenspace of the Laplacian acting on Dirichlet functions is 1-dimensional.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer-Verlag</pub><doi>10.1007/s00220-010-1185-6</doi><tpages>54</tpages><orcidid>https://orcid.org/0009-0009-6010-151X</orcidid></addata></record> |
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| ispartof | Communications in mathematical physics, 2011, Vol.302 (2), p.291-344 |
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| language | eng |
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| source | Springer Nature Link eJournals |
| subjects | Algebra Analysis of PDEs Classical and Quantum Gravitation Complex Systems Differential geometry Exact sciences and technology Geometry Mathematical and Computational Physics Mathematical methods in physics Mathematical Physics Mathematics Number theory Other topics in mathematical methods in physics Physics Physics and Astronomy Quantum Physics Relativity Theory Sciences and techniques of general use Spectral Theory Theoretical |
| title | Spectral Simplicity and Asymptotic Separation of Variables |
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