On Spectral Problems of Discrete Schrödinger Operators

Applications of Mathematics (2021)Cite this article

Abstract

A special type of Jacobi matrices, discrete Schrödinger operators, is found to play an important role in quantum physics. In this paper, we show that given the spectrum of a discrete Schrödinger operator and the spectrum of the operator obtained by deleting the first row and the first column of it can determine the discrete Schrödinger operator uniquely, even though one eigenvalue of the latter is missing. Moreover, we find the forms of the discrete Schrödinger operators when their smallest and largest eigenvalues attain the extrema under certain constraints by use of the notion of generalized directional derivative and the method of Lagrange multiplier.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

US$ 39.95

Tax calculation will be finalised during checkout.

References

  1. [1]

    Z. S. Agranovich, V. A. Marchenko: The Inverse Problem of Scattering Theory. Gordon and Breach, New York, 1963.

    Google Scholar 

  2. [2]

    P. W. Anderson: Absence of diffusion in certain random lattices. Phys. Rev. 109 (1958), 1492–1505.

    Article  Google Scholar 

  3. [3]

    A. Astrauskas: Extremal theory for spectrum of random discrete Schrödinger operator. III. Localization properties. J. Stat. Phys. 150 (2013), 889–907.

    MathSciNet  Article  Google Scholar 

  4. [4]

    G. Borg: Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe: Bestimmung der Differentialgleichung durch die Eigenwerte. Acta Math. 78 (1946), 1–96. (In German.)

    MathSciNet  Article  Google Scholar 

  5. [5]

    D. Damanik, D. Hundertmark, R. Killip, B. Simon: Variational estimates for discrete Schrödinger operators with potentials of indefinite sign. Commun. Math. Phys. 238 (2003), 545–562.

    Article  Google Scholar 

  6. [6]

    I. M. Gel’fand, B. M. Levitan: On the determination of a differential equation from its spectral function. Am. Math. Soc., Transl., II. Ser. 1 (1955), 253–304.

    MathSciNet  Article  Google Scholar 

  7. [7]

    F. Gesztesy, B. Simon: m-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices. J. Anal. Math. 73 (1997), 267–297.

    MathSciNet  Article  Google Scholar 

  8. [8]

    G. M. L. Gladwell: Inverse Problems in Vibrations. Mechanics: Dynamical Systems 9. Martinus Nijhoff Publishers, Dordrecht, 1986.

    Google Scholar 

  9. [9]

    G. H. Hardy, J. E. Littlewood, G. Pólya: Inequalities. Cambridge University Press, Cambridge, 1952.

    Google Scholar 

  10. [10]

    H. Hochstadt: On some inverse problems in matrix theory. Arch. Math. 18 (1967), 201–207.

    MathSciNet  Article  Google Scholar 

  11. [11]

    H. Hochstadt: The inverse Sturm-Liouville problem. Commun. Pure Appl. Math. 26 (1973), 715–729.

    MathSciNet  Article  Google Scholar 

  12. [12]

    N. Levinson: The inverse Sturm-Liouville problem. Mat. Tidsskr. B 1949 (1949), 25–30.

    MathSciNet  MATH  Google Scholar 

  13. [13]

    B. M. Levitan: Inverse Sturm-Liouville Problem. VNU Science Press, Utrecht, 1987.

    Google Scholar 

  14. [14]

    B. M. Levitan, M. G. Gasymov: Determination of a differential equation by two of its spectra. Russ. Math. Surv. 19 (1964), 1–63; translation from Usp. Mat. Nauk 19 (1964), 1–63.

    Article  Google Scholar 

  15. [15]

    D. E. Pelinovsky, A. Stefanov: On the spectral theory and dispersive estimates for a discrete Schrödinger equation in one dimension. J. Math. Phys. 49 (2008), Article ID 113501, 17 pages.

  16. [16]

    J. Pöschel, E. Trubowitz: Inverse Spectral Theory. Pure and Applied Mathematics 130. Academic Press, Boston, 1987.

    Google Scholar 

Download references

Author information

Affiliations

  1. Department of Mathematics, National Tsing Hua University, Hsinchu, 30013, Taiwan

    Chi-Hua Chan

  2. Department of Mathematics and Information Education, National Taipei University of Education, Taipei, 10671, Taiwan

    Po-Chun Huang

Authors
  1. Chi-Hua Chan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Po-Chun Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chi-Hua Chan.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chan, CH., Huang, PC. On Spectral Problems of Discrete Schrödinger Operators. Appl Math (2021). https://doi.org/10.21136/AM.2021.0203-19

Download citation

Keywords

  • discrete Schrödinger operator

MSC 2020

  • 34B09