Communications in Mathematical Physics

, Volume 359, Issue 2, pp 499–514 | Cite as

Reflectionless Discrete Schrödinger Operators are Spectrally Atypical

Article
  • 40 Downloads

Abstract

We prove that, if an isospectral torus contains a discrete Schrödinger operator with nonconstant potential, the shift dynamics on that torus cannot be minimal. Consequently, we specify a generic sense in which finite unions of nondegenerate closed intervals having capacity one are not the spectrum of any reflectionless discrete Schrödinger operator. We also show that the only reflectionless discrete Schrödinger operators having zero, one, or two spectral gaps are periodic.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Avila A.: On the Kotani-Last and Schrödinger conjectures. J. Am. Math. Soc. 28(2), 579–616 (2015)CrossRefMATHGoogle Scholar
  2. 2.
    Borg G.: Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte. Acta Math. 78, 1–96 (1946)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Breuer J., Ryckman E., Simon B.: Equality of the spectral and dynamical definitions of reflection. Commun. Math. Phys. 295(2), 531–550 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Damanik, D., Goldstein, M., Schlag, W., Voda, M.: Homogeneity of the spectrum for quasi-perioidic Schrödinger operators. (2015). arXiv preprint arXiv:1505.04904
  5. 5.
    Denisov S.: On Rakhmanov’s theorem for Jacobi matrices. Proc. Am. Math. Soc. 132(3), 847–852 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Eremenko, A., Yuditskii, P.: Comb functions. In: Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications, Volume 578 of Contemporary Mathematics, pp. 99–118. American Mathematical Society, Providence (2012)Google Scholar
  7. 7.
    Fillman, J., Lukic, M.: Spectral homogeneity of limit-periodic Schrödinger operators. J. Spectr. Theory 7(2), 387–406 (2017)Google Scholar
  8. 8.
    Gesztesy, F., Holden, H., Michor, J., Teschl, G.: Soliton Equations and Their Algebro-Geometric Solutions. Vol. II, (1 + 1)-Dimensional Discrete Models, Volume 114 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2008)Google Scholar
  9. 9.
    Gesztesy F., Makarov K.A., Zinchenko M.: Essential closures and AC spectra for reflectionless CMV, Jacobi, and Schrödinger operators revisited. Acta Appl. Math. 103(3), 315–339 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gesztesy F., Yuditskii P.: Spectral properties of a class of reflectionless Schrödinger operators. J. Funct. Anal. 241(2), 486–527 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gesztesy F., Zinchenko M.: Local spectral properties of reflectionless Jacobi, CMV, and Schrödinger operators. J. Differ. Equ. 246(1), 78–107 (2009)ADSCrossRefMATHGoogle Scholar
  12. 12.
    Hochstadt H.: On the theory of Hill’s matrices and related inverse spectral problems. Linear Algebra Appl. 11, 41–52 (1975)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hur, I.: The m-functions of discrete Schrödinger operators are sparse compared to those for Jacobi operators. (2017). arXiv preprint arXiv:1703.03494
  14. 14.
    Killip R., Simon B.: Sum rules for Jacobi matrices and their applications to spectral theory. Ann. Math. (2) 158(1), 253–321 (2003)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kotani, S.: Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators. In: Stochastic Analysis (Katata/Kyoto, 1982), Volume 32 of North-Holland Math. Library, pp. 225–247. North-Holland, Amsterdam (1984)Google Scholar
  16. 16.
    Poltoratski, A., Remling, C., : Approximation results for reflectionless Jacobi matrices. Int. Math. Res. Not. 2011 16, 3575–3617 (2011)Google Scholar
  17. 17.
    Ponomarëv S.: Submersions and pre-images of sets of measure zero. Sibirsk. Mat. Zh. 28(1), 199–210 (1987)MathSciNetGoogle Scholar
  18. 18.
    Pöschel J.: Examples of discrete Schrödinger operators with pure point spectrum. Comm. Math. Phys. 88(4), 447–463 (1983)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ransford T.: Potential Theory in the Complex Plane, Volume 28 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  20. 20.
    Remling C.: The absolutely continuous spectrum of Jacobi matrices. Ann. Math. (2) 174(1), 125–171 (2011)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Remling C.: Topological properties of reflectionless Jacobi matrices. J. Approx. Theory 168, 1–17 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Shen J., Strang G., Wathen A.: The potential theory of several intervals and its applications. Appl. Math. Optim. 44(1), 67–85 (2001)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Simon, B.: Szegő’s theorem and its descendants. Spectral theory for L 2 perturbations of orthogonal polynomials. M. B. Porter Lectures. Princeton University Press, Princeton (2011)Google Scholar
  24. 24.
    Sodin M., Yuditskii P.: Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions. J. Geom. Anal. 7(3), 387–435 (1997)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Teschl G.: Jacobi Operators and Completely Integrable Nonlinear Lattices, Volume 72 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2000)Google Scholar
  26. 26.
    Totik V.: Chebyshev constants and the inheritance problem. J. Approx. Theory 160, 187–201 (2009)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Vinnikov V., Yuditskii P.: Functional models for almost periodic Jacobi matrices and the Toda hierarchy. Mat. Fiz. Anal. Geom. 9(2), 206–219 (2002)MathSciNetMATHGoogle Scholar
  28. 28.
    Volberg A., Yuditskii P.: Kotani-Last problem and Hardy spaces on surfaces of Widom type. Invent. Math. 197(3), 683–740 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Widom H.: Extremal polynomials associated with a system of curves in the complex plane. Adv. Math. 3, 127–232 (1969)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsRice UniversityHoustonUSA

Personalised recommendations