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Eigenfunctions of Unbounded Support for Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators

Abstract

It is known that if a locally perturbed periodic self-adjoint operator on a combinatorial or quantum graph admits an eigenvalue embedded in the continuous spectrum, then the associated eigenfunction is compactly supported—that is, if the Fermi surface is irreducible, which occurs generically in dimension two or higher. This article constructs a class of operators whose Fermi surface is reducible for all energies by coupling several periodic systems. The components of the Fermi surface correspond to decoupled spaces of hybrid states, and in certain frequency bands, some components contribute oscillatory hybrid states (corresponding to spectrum) and other components contribute only exponential ones. This separation allows a localized defect to suppress the oscillatory (radiation) modes and retain the exponential ones, thereby leading to embedded eigenvalues whose associated eigenfunctions decay exponentially but are not compactly supported.

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References

  1. 1

    Hugo A., Ricardo C., Peter Z.: Scattering and embedded trapped modes for an infinite nonhomogeneous Timoshenko beam. Kluwer, Dordrecht (2012)

    Google Scholar 

  2. 2

    Bättig D., Knörrer H., Trubowitz E.: A directional compactification of the complex Fermi surface. Composit. Math. 79(2), 205–229 (1991)

    MATH  Google Scholar 

  3. 3

    Berkolaiko, G., Kuchment, P.: Introduction to quantum graphs. Mathematical Surveys and Monographs, vol. 186. AMS, Providence (2013)

  4. 4

    Gieseker D., Knörrer H., Trubowitz E.: The Geometry of Algebraic Fermi Curves. Academic Press, Boston (1993)

    MATH  Google Scholar 

  5. 5

    Knörrer H., Trubowitz E.: A directional compactification of the complex Bloch variety. Comment. Math. Helv. 65, 114–149 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6

    Peter K.: Floquet Theory for Partial Differential Equations. Birkhäuser Verlag AG, Basel (1993)

    MATH  Google Scholar 

  7. 7

    Peter K.: Quantum graphs II. some spectral properties of quantum and combinatorial graphs. J. Phys. A 38, 4887–4900 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. 8

    Peter K., Boris V.: On absence of embedded eigenvalues for Schrödinger operators with perturbed periodic potentials. Commun. Part. Differ. Equ. 25(9–10), 1809–1826 (2000)

    MATH  Google Scholar 

  9. 9

    Peter K., Boris V.: On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators. Commun. Math. Phys. 268(3), 673–686 (2006)

    Article  MATH  Google Scholar 

  10. 10

    Jeffrey S.H., Michael A.: The creation of spectral gaps by graph decoration. Lett. Math. Phys. 53(3), 253–262 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. 11

    Stephen S.P., Jennifer R., Katherine S.H., Clayton W.: A discrete model for resonance near embedded bound states. IEEE Photonics J. 2(6), 911–923 (2010)

    Article  Google Scholar 

  12. 12

    Shipman, S.P., Welters, A.T.: Resonant electromagnetic scattering in anisotropic layered media. J. Math. Phys. 54(10), 103511–1–40 (2013)

  13. 13

    Tillay, J.: Resonance between bound states and radiation in lattices, Undergraduate poster, Louisiana State University (2012). https://www.math.lsu.edu/~shipman/WebDocuments/Tillay2012.pdf

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Correspondence to Stephen P. Shipman.

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Communicated by P. Deift

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Shipman, S.P. Eigenfunctions of Unbounded Support for Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators. Commun. Math. Phys. 332, 605–626 (2014). https://doi.org/10.1007/s00220-014-2113-y

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Keywords

  • Fermi Surface
  • Fundamental Domain
  • Hybrid State
  • Laurent Polynomial
  • Quantum Graph