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Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map

  • Gregory Berkolaiko
  • Graham CoxEmail author
  • Jeremy L. Marzuola
Article
  • 19 Downloads

Abstract

It has been recently shown that the nodal deficiency of an eigenfunction is encoded in the spectrum of the Dirichlet-to-Neumann operators for the eigenfunction’s positive and negative nodal domains. While originally derived using symplectic methods, this result can also be understood through the spectral flow for a family of boundary conditions imposed on the nodal set, or, equivalently, a family of operators with delta function potentials supported on the nodal set. In this paper, we explicitly describe this flow for a Schrödinger operator with separable potential on a rectangular domain and determine a mechanism by which lower-energy eigenfunctions do or do not contribute to the nodal deficiency.

Keywords

Nodal domains Nodal sets Dirichlet-to-Neumann map Spectral indices 

Mathematics Subject Classification

58J50 35B05 35P05 35J10 

Notes

Acknowledgements

The authors would like to thank Ram Band for interesting discussions and helpful suggestions regarding the manuscript. G.B. acknowledges partial support from the NSF under Grant DMS-1410657. G.C. acknowledges the support of NSERC Grant RGPIN-2017-04259. J.L.M. was supported in part by NSF Applied Math Grant DMS-1312874 and NSF CAREER Grant DMS-1352353.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA
  2. 2.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’sCanada
  3. 3.Department of MathematicsUniversity of North Carolina at Chapel HillChapel HillUSA

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