Limiting eigenfunctions of Sturm–Liouville operators subject to a spectral flow
We examine the spectrum of a family of Sturm–Liouville operators with regularly spaced delta function potentials parametrized by increasing strength. The limiting behavior of the eigenvalues under this spectral flow was described by Berkolaiko et al. (Lett Math Phys 109(7):1611–1623, 2019), where it...
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| Published in | Annales mathématiques du Québec Vol. 45; no. 2; pp. 249 - 269 |
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| Main Authors | , , , , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.10.2021
Springer Nature B.V |
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| ISSN | 2195-4755 2195-4763 |
| DOI | 10.1007/s40316-020-00142-6 |
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| Abstract | We examine the spectrum of a family of Sturm–Liouville operators with regularly spaced delta function potentials parametrized by increasing strength. The limiting behavior of the eigenvalues under this spectral flow was described by Berkolaiko et al. (Lett Math Phys 109(7):1611–1623, 2019), where it was used to study the nodal deficiency of Laplacian eigenfunctions. Here we consider the eigenfunctions of these operators. In particular, we give explicit formulas for the limiting eigenfunctions, and also characterize the eigenfunctions and eigenvalues for all values for the spectral flow parameter (not just in the limit). We also develop spectrally accurate numerical tools for comparison and visualization. |
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| AbstractList | We examine the spectrum of a family of Sturm–Liouville operators with regularly spaced delta function potentials parametrized by increasing strength. The limiting behavior of the eigenvalues under this spectral flow was described by Berkolaiko et al. (Lett Math Phys 109(7):1611–1623, 2019), where it was used to study the nodal deficiency of Laplacian eigenfunctions. Here we consider the eigenfunctions of these operators. In particular, we give explicit formulas for the limiting eigenfunctions, and also characterize the eigenfunctions and eigenvalues for all values for the spectral flow parameter (not just in the limit). We also develop spectrally accurate numerical tools for comparison and visualization. |
| Author | Cox, Graham Bors, Isabel Marzuola, Jeremy L. Beck, Thomas Conte, Grace |
| Author_xml | – sequence: 1 givenname: Thomas surname: Beck fullname: Beck, Thomas organization: Department of Mathematics, University of North Carolina at Chapel Hill – sequence: 2 givenname: Isabel surname: Bors fullname: Bors, Isabel organization: Department of Mathematics, University of North Carolina at Chapel Hill – sequence: 3 givenname: Grace surname: Conte fullname: Conte, Grace organization: Department of Mathematics, University of North Carolina at Chapel Hill – sequence: 4 givenname: Graham surname: Cox fullname: Cox, Graham organization: Department of Mathematics and Statistics, Memorial University of Newfoundland – sequence: 5 givenname: Jeremy L. orcidid: 0000-0002-8668-7143 surname: Marzuola fullname: Marzuola, Jeremy L. email: marzuola@math.unc.edu organization: Department of Mathematics, University of North Carolina at Chapel Hill |
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| Cites_doi | 10.1090/proc/13098 10.1007/s10543-008-0198-4 10.1002/cpa.3160090324 10.1007/s00039-012-0199-y 10.1016/j.crma.2015.03.014 10.1016/j.anihpc.2007.07.004 10.1093/imanum/drv013 10.3934/dcds.2019093 10.1007/s11005-019-01159-x 10.1090/conm/700/14182 10.1007/s11005-016-0819-9 10.1512/iumj.2017.66.6129 10.4171/PM/2059 10.1007/978-1-4419-1345-6_6 10.1515/9783110550887-010 |
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| Keywords | 34B24 35P05 Nodal partitions 34C10 34L05 Spectral flows Nodal deficiency Sturm-Liouville operators |
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| References | CR4 CR6 Léna (CR14) 2015; 353 Bérard, Helffer (CR3) 2016; 106 Kuan, Hale (CR16) 2016; 36 Cox, Jones, Marzuola (CR7) 2017; 66 Helffer, Hoffmann-Ostenhof, Terracini (CR11) 2009; 26 Goodman (CR10) 2019; 39 CR12 Pleijel (CR15) 1956; 9 Driscoll, Hale (CR9) 2016; 36 Berkolaiko (CR2) 2017; 700 Driscoll, Bornemann, Trefethen (CR8) 2008; 48 Helffer, Sundqvist (CR13) 2016; 144 Berkolaiko, Cox, Marzuola (CR1) 2019; 109 Berkolaiko, Kuchment, Smilansky (CR5) 2012; 22 TA Driscoll (142_CR8) 2008; 48 142_CR12 B Helffer (142_CR13) 2016; 144 B Helffer (142_CR11) 2009; 26 142_CR6 G Berkolaiko (142_CR2) 2017; 700 142_CR4 G Berkolaiko (142_CR1) 2019; 109 Å Pleijel (142_CR15) 1956; 9 X Kuan (142_CR16) 2016; 36 G Berkolaiko (142_CR5) 2012; 22 C Léna (142_CR14) 2015; 353 RH Goodman (142_CR10) 2019; 39 TA Driscoll (142_CR9) 2016; 36 P Bérard (142_CR3) 2016; 106 G Cox (142_CR7) 2017; 66 |
| References_xml | – volume: 144 start-page: 4777 issue: 11 year: 2016 end-page: 4791 ident: CR13 article-title: On nodal domains in Euclidean balls publication-title: Proc. Am. Math. Soc. doi: 10.1090/proc/13098 – ident: CR4 – volume: 48 start-page: 701 issue: 4 year: 2008 end-page: 723 ident: CR8 article-title: The Chebop system for automatic solution of differential equations publication-title: BIT Numer. Math. doi: 10.1007/s10543-008-0198-4 – ident: CR12 – volume: 9 start-page: 543 year: 1956 end-page: 550 ident: CR15 article-title: Remarks on Courant’s nodal line theorem publication-title: Commun. Pure Appl. Math. doi: 10.1002/cpa.3160090324 – volume: 22 start-page: 1517 issue: 6 year: 2012 end-page: 1540 ident: CR5 article-title: Critical partitions and nodal deficiency of billiard eigenfunctions publication-title: Geom. Funct. Anal. doi: 10.1007/s00039-012-0199-y – volume: 353 start-page: 535 issue: 6 year: 2015 end-page: 539 ident: CR14 article-title: Courant-sharp eigenvalues of a two-dimensional torus publication-title: C.R. Math. doi: 10.1016/j.crma.2015.03.014 – ident: CR6 – volume: 26 start-page: 101 year: 2009 end-page: 138 ident: CR11 article-title: Nodal domains and spectral minimal partitions publication-title: Ann. de l’IHP Anal. Non Linéaire doi: 10.1016/j.anihpc.2007.07.004 – volume: 36 start-page: 108 issue: 1 year: 2016 end-page: 132 ident: CR9 article-title: Rectangular spectral collocation publication-title: IMA J. Numer. Anal. – volume: 36 start-page: 618 issue: 2 year: 2016 end-page: 632 ident: CR16 article-title: Explicit construction of rectangular differentiation matrices publication-title: IMA J. Numer. Anal. doi: 10.1093/imanum/drv013 – volume: 39 start-page: 2203 issue: 4 year: 2019 end-page: 2232 ident: CR10 article-title: NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph publication-title: Discrete Contin. Dyn. Syst. doi: 10.3934/dcds.2019093 – volume: 109 start-page: 1611 issue: 7 year: 2019 end-page: 1623 ident: CR1 article-title: Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map publication-title: Lett. Math. Phys. doi: 10.1007/s11005-019-01159-x – volume: 700 start-page: 41 year: 2017 end-page: 72 ident: CR2 article-title: An elementary introduction to quantum graphs publication-title: Geom. Comput. Spectr. Theory doi: 10.1090/conm/700/14182 – volume: 106 start-page: 1729 issue: 12 year: 2016 end-page: 1789 ident: CR3 article-title: Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle publication-title: Lett. Math. Phys. doi: 10.1007/s11005-016-0819-9 – volume: 66 start-page: 1573 issue: 5 year: 2017 end-page: 1602 ident: CR7 article-title: Manifold decompositions and indices of Schrödinger operators publication-title: Indiana Univ. Math. J. doi: 10.1512/iumj.2017.66.6129 – volume: 22 start-page: 1517 issue: 6 year: 2012 ident: 142_CR5 publication-title: Geom. Funct. Anal. doi: 10.1007/s00039-012-0199-y – ident: 142_CR4 doi: 10.4171/PM/2059 – volume: 39 start-page: 2203 issue: 4 year: 2019 ident: 142_CR10 publication-title: Discrete Contin. Dyn. Syst. doi: 10.3934/dcds.2019093 – volume: 9 start-page: 543 year: 1956 ident: 142_CR15 publication-title: Commun. Pure Appl. Math. doi: 10.1002/cpa.3160090324 – volume: 48 start-page: 701 issue: 4 year: 2008 ident: 142_CR8 publication-title: BIT Numer. Math. doi: 10.1007/s10543-008-0198-4 – volume: 106 start-page: 1729 issue: 12 year: 2016 ident: 142_CR3 publication-title: Lett. Math. Phys. doi: 10.1007/s11005-016-0819-9 – volume: 26 start-page: 101 year: 2009 ident: 142_CR11 publication-title: Ann. de l’IHP Anal. Non Linéaire doi: 10.1016/j.anihpc.2007.07.004 – volume: 36 start-page: 108 issue: 1 year: 2016 ident: 142_CR9 publication-title: IMA J. Numer. Anal. – volume: 109 start-page: 1611 issue: 7 year: 2019 ident: 142_CR1 publication-title: Lett. Math. Phys. doi: 10.1007/s11005-019-01159-x – volume: 36 start-page: 618 issue: 2 year: 2016 ident: 142_CR16 publication-title: IMA J. Numer. Anal. doi: 10.1093/imanum/drv013 – volume: 144 start-page: 4777 issue: 11 year: 2016 ident: 142_CR13 publication-title: Proc. Am. Math. Soc. doi: 10.1090/proc/13098 – volume: 353 start-page: 535 issue: 6 year: 2015 ident: 142_CR14 publication-title: C.R. Math. doi: 10.1016/j.crma.2015.03.014 – volume: 700 start-page: 41 year: 2017 ident: 142_CR2 publication-title: Geom. Comput. Spectr. Theory doi: 10.1090/conm/700/14182 – ident: 142_CR12 doi: 10.1007/978-1-4419-1345-6_6 – ident: 142_CR6 doi: 10.1515/9783110550887-010 – volume: 66 start-page: 1573 issue: 5 year: 2017 ident: 142_CR7 publication-title: Indiana Univ. Math. J. doi: 10.1512/iumj.2017.66.6129 |
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| Snippet | We examine the spectrum of a family of Sturm–Liouville operators with regularly spaced delta function potentials parametrized by increasing strength. The... |
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| SubjectTerms | Algebra Analysis Constraining Delta function Eigenvalues Eigenvectors Mathematics Mathematics and Statistics Number Theory Operators (mathematics) Spectra |
| Title | Limiting eigenfunctions of Sturm–Liouville operators subject to a spectral flow |
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