Skip to main content
SpringerLink
Log in

The xi function

  • Published:
Acta Mathematica

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Atkinson, F. V., On the location of the Weyl circles.Proc. Roy. Soc. Edinburgh Sect. A, 88 (1981), 345–356.

    MathSciNet  Google Scholar 

  2. Avron, J., Simon, B. &Mouche, P. van, On the measure of the spectrum for the almost Mathieu operator.Comm. Math. Phys., 132 (1990), 103–118.

    Article  MathSciNet  Google Scholar 

  3. Borg, G. Uniqueness theorems in the spectral theory ofy"+(λ−q(x))y=0 inProc. 11th Scandinavian Congress of Mathematicians, pp. 276–287. Johan Grundt Tanums Forlag, Oslo, 1952.

    Google Scholar 

  4. Chulaevsky, V., On perturbations of a Schrödinger operator with periodic potential.Russian Math. Surveys, 36:5 (1981), 143–144.

    Article  Google Scholar 

  5. Craig, W. The trace formula for Schrödinger operators on the line.Comm. Math. Phys., 126 (1989), 379–407.

    Article  MathSciNet  Google Scholar 

  6. Deift, P &Simon, B. Almost periodic Schrödinger operators. III. The absolutely continuous spectrum in one dimension.Comm. Math. Phys. 90 (1983), 389–411.

    Article  MathSciNet  Google Scholar 

  7. Deift, P &Trubowitz, E. Inverse scattering on the line.Comm. Pure Appl. Math., 32 (1979), 121–251.

    MathSciNet  Google Scholar 

  8. Dikii, L. A., Trace formulas for Sturm-Liouville differential operators.Amer. Math. Soc. Transl. Ser. 2, 18 (1961), 81–115.

    MathSciNet  Google Scholar 

  9. Dubrovin, B. A., Periodic problems for the Korteweg-de Vries equation in the class of finite band potentials.Functional Anal. Appl., 9 (1975), 215–223.

    Article  Google Scholar 

  10. Eastham, M. S. P.,The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburgh, 1973.

    MATH  Google Scholar 

  11. Everitt, W. N., On a property of them-coefficient of a second-order linear differential equation.J. London Math. Soc., 4 (1972), 443–457.

    MathSciNet  Google Scholar 

  12. Flaschka, H., On the inverse problem for Hill's operator.Arch. Rational Mech. Anal., 59 (1975), 293–309.

    Article  MathSciNet  Google Scholar 

  13. Gel'fand, I. M., On identities for eigenvalues of a second order differential operator.Uspekhi Mat. Nauk, 11:1 (1956), 191–198 (Russian); English translation inIzrail M. Gelfand, Collected Papers, Vol. I (S. G. Gindikin, V. W. Guillemin, A. A. Kirillov, B. Kostant, S. Sternberg, eds.), pp. 510–517. Springer-Verlag, Berlin, 1987.

    Google Scholar 

  14. Gel'fand, I. M. &Levitan, B. M., On the determination of a differential equation from its spectral function.Izv. Akad. Nauk SSSR Ser. Mat., 15 (1951), 309–360 (Russian); English translation inAmer. Math. Soc. Transl. Ser. 2, 1 (1955), 253–304.

    MathSciNet  Google Scholar 

  15. —, On a simple identity for eigenvalues of a second order differential operator.Dokl. Akad. Nauk SSSR, 88 (1953), 593–596 (Russian); English translation inIzrail M. Gelfand, Collected Papers, Vol. I (S. G. Gindikin, V. W. Guillemin, A. A. Kirillov, B. Kostant, S. Sternberg, eds.), pp. 457–461. Springer-Verlag, Berlin, 1987.

    Google Scholar 

  16. Gesztesy, F., Holden, H. &Simon, B., Absolute summability of the trace relation for certain Schrödinger operators.Comm. Math. Phys., 168 (1995), 137–161.

    Article  MathSciNet  Google Scholar 

  17. Gesztesy, F., Holden, H., Simon, B. &Zhao, Z., Higher order trace relations for Schrödinger operators.Rev. Math. Phys., 7 (1995), 893–922.

    Article  MathSciNet  Google Scholar 

  18. Gesztesy, F., A trace formula for multidimensional Schrödinger operators. To appear inJ. Funct. Anal.

  19. Gesztesy F. &Simon, B. Rank one perturbations at infinite coupling.J. Funct. Anal., 128 (1995), 245–252.

    Article  MathSciNet  Google Scholar 

  20. Gesztesy, F. Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators. To appear inTrans. Amer. Math. Soc.

  21. Gesztesy, F, Krishna, M. & Teschl, G., On isospectral sets of Jacobi operators. Preprint, 1996.

  22. Gilbert, R. C. &Kramer, V. A., Trace formulas for powers of a Sturm-Liouville operator.Canad. J. Math., 16 (1964), 412–422.

    MathSciNet  Google Scholar 

  23. Halberg, C. J. A. &Kramer, V. A., A generalization of the trace concept.Duke Math. J., 27 (1960), 607–617.

    Article  MathSciNet  Google Scholar 

  24. Hochstadt, H., On the determination of a Hill's equation from its spectrum.Arch. Rational Mech. Anal., 19 (1965), 353–362.

    Article  MathSciNet  Google Scholar 

  25. Iwasaki, K. Inverse problem for Sturm-Liouville and Hill equations.Ann. Mat. Pura Appl. (4), 149 (1987), 185–206.

    Article  MathSciNet  Google Scholar 

  26. Johnson, R. &Moser, J., The rotation number for almost periodic potentials.Comm. Math. Phys., 84 (1982), 403–438.

    Article  MathSciNet  Google Scholar 

  27. Kato, T. Perturbation Theory for Linear Operators, 2nd edition, Springer-Verlag, Berlin, 1980.

    MATH  Google Scholar 

  28. Katznelson, Y.,An Introduction to Harmonic Analysis, Dover, New York, 1976.

    MATH  Google Scholar 

  29. Knill, O. & Last, Y., Spectral properties of Jacobi matrices arising from twist maps. In preparation.

  30. Kotani, S., Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators, inStochastic Analysis (K. Ito, ed.), pp. 225–247. North-Holland, Amsterdam, 1984.

    Google Scholar 

  31. Kotani, S &Krishna, M., Almost periodicity of some random potentials.J. Funct. Anal., 78 (1988), 390–405.

    Article  MathSciNet  Google Scholar 

  32. Krein, M. G., Perturbation determinants and a formula for the traces of unitary and self-adjoint operators.Soviet. Math. Dokl., 3 (1962), 707–710.

    Google Scholar 

  33. Last, Y. A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants.Comm. Math. Phys., 151 (1993), 183–192.

    Article  MathSciNet  Google Scholar 

  34. Levitan, B. M., On the closure of the set of finite-zone potentials.Math. USSR-Sb., 51 (1985), 67–89.

    Article  Google Scholar 

  35. —,Inverse Sturm-Liouville Problems, VNU Science Press, Utrecht, 1987.

    MATH  Google Scholar 

  36. Levitan, B. M. &Gasymov, M. G., Determination of a differential equation by two of its spectra.Russian Math. Surveys, 19:2 (1964), 1–63.

    Article  MathSciNet  Google Scholar 

  37. Magnus, W. &Winkler, S.,Hill's Equation, Dover, New York, 1979.

    Google Scholar 

  38. Marchenko, V. A., Some questions in the theory of one-dimensional linear differential operators of the second order, I,Trudy Moskov. Mat. Obshch., 1 (1952), 327–420 (Russian); English translation inAmer. Math. Soc. Transl. Ser. 2, 101 (1973), 1–104.

    Google Scholar 

  39. —Sturm-Liouville Operators and Applications, Birkhäuser, Basel, 1986.

    MATH  Google Scholar 

  40. Marchenko, V. A. &Ostrovskii, I. V., A characterization of the spectrum of Hill's operator.Math. USSR-Sb., 26 (1975), 493–554.

    Article  Google Scholar 

  41. McKean, H. P. &Moerbeke, P. van, The spectrum of Hill's equation.Invent. Math., 30 (1975), 217–274.

    Article  MathSciNet  Google Scholar 

  42. McKean, H. P. &Trubowitz, E., Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points.Comm. Pure Appl. Math., 29 (1976), 143–226.

    MathSciNet  Google Scholar 

  43. Reed, M. &Simon, B.,Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975.

    MATH  Google Scholar 

  44. —Methods of Modern Mathematical Physics, IV.Analysis of Operators, Academic Press, New York, 1978.

    Google Scholar 

  45. Simon, B., A canonical decomposition for quadratic forms with applications to monotone convergence theorems.J. Funct. Anal., 28 (1978), 377–385.

    Article  MathSciNet  Google Scholar 

  46. —,Functional Integration and Quantum Physics Academic Press, New York, 1979.

    Google Scholar 

  47. —,Trace Ideals and their Applications, Cambridge Univ. Press, Cambridge, 1979.

    Google Scholar 

  48. —, Spectral analysis of rank one perturbations and applications, inProc. Mathematical Quantum Theory, II: Schrödinger Operators (J. Feldman, R. Froese and L. Rosen, eds.), pp. 109–149. CRM Proc. Lecture Notes, 8, Amer. Math. Soc., Providence, RI, 1995.

    Google Scholar 

  49. Trubowitz, E., The inverse problem for periodic potentials.Comm. Pure Appl. Math., 30 (1977), 321–337.

    MathSciNet  Google Scholar 

  50. Venakides, S., The infinite period limit of the inverse formalism for periodic potentials.Comm. Pure Appl. Math., 41 (1988), 3–17.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Missouri, 65211, Columbia, MO, USA

    Fritz Gesztesy

  2. Division of Physics, Mathematics, and Astronomy, California Institute of Technology, 253-37, Pasadena, CA, 91125, USA

    Barry Simon

Authors
  1. Fritz Gesztesy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Barry Simon
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This material is based upon work supported by the National Science Foundation under Grant DMS-9101715. The Government has certain rights in this material.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gesztesy, F., Simon, B. The xi function. Acta Math. 176, 49–71 (1996). https://doi.org/10.1007/BF02547335

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02547335

Keywords

Search

Navigation