Skip to main content
SpringerLink
Log in

The Ξ operator and its relation to Krein's spectral shift function

  • Published:
Journal d’Analyse Mathématique Aims and scope

Abstract

We explore connections between Krein's spectral shift function ζ(λ,H 0, H) associated with the pair of self-adjoint operators (H 0, H),H=H 0+V, in a Hilbert spaceH and the recently introduced concept of a spectral shift operator Ξ(J+K *(H 0−λ−i0)−1 K) associated with the operator-valued Herglotz functionJ+K *(H 0−z)−1 K, Im(z)>0 inH, whereV=KJK * andJ=sgn(V). Our principal results include a new representation for ζ(λ,H 0,H) in terms of an averaged index for the Fredholm pair of self-adjoint spectral projections (E J+A(λ)+tB(λ)(−∞, 0)),E J((−∞, 0))), ℝ, whereA(λ)=Re(K *(H 0−λ−i0−1 K),B(λ)=Im(K *(H 0−λ-i0)−1 K) a.e. Moreover, introducing the new concept of a trindex for a pair of operators (A, P) inH, whereA is bounded andP is an orthogonal projection, we prove that ζ(λ,H 0, H) coincides with the trindex associated with the pair (Ξ(J+K *(H 0−λ−i0)K), Ξ(J)). In addition, we discuss a variant of the Birman-Krein formula relating the trindex of a pair of Ξ operators and the Fredholm determinant of the abstract scattering matrix.

We also provide a generalization of the classical Birman—Schwinger principle, replacing the traditional eigenvalue counting functions by appropriate spectral shift functions.

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

Similar content being viewed by others

References

  1. W. O. Amrein and K. B. Sinha,On pairs of projections in a Hilbert space, Linear Algebra Appl.208/209 (1994), 425–435.

    Article  MathSciNet  Google Scholar 

  2. N. Aronszajn and W. F. Donoghue,On exponential representations of analytic functions in the upper half-plane with positive imaginary part J. Analyse Math.5 (1956-57), 321–388.

    Article  Google Scholar 

  3. J. Avron, R. Seiler and B. Simon,The index of a pair of projections, J. Funct. Anal.120 (1994), 220–237.

    Article  MATH  MathSciNet  Google Scholar 

  4. H. Baumgärtel and M. Wollenberg,Mathematical Scattering Theory, Birkhäuser, Basel, 1983.

    Google Scholar 

  5. M. Š. Birman,The spectrum of singular boundary problems, Mat. Sb. (N.S.)55 (1961), 125–174 (Russian); Engl. transl.: Amer. Math. Soc. Transl. (2)53 (1966), 23–80.

    MathSciNet  Google Scholar 

  6. M. Sh. Birman and S. B. Entina,The stationary method in the abstract theory of scattering, Math. USSR Izv.1 (1967), 391–420.

    Article  Google Scholar 

  7. M. Sh. Birman and M. G. Krein,On the theory of wave operators and scattering operators, Sov. Math. Dokl.3 (1962), 740–744.

    MATH  Google Scholar 

  8. M. Sh. Birman and A. B. Pushnitski,Spectral shift function, amazing and multifaceted, Integral Equations Operator Theory30 (1998), 191–199.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Sh. Birman and M. Z. Solomyak,Remarks on the spectral shift function, J. Sov. Math.3 (1975), 408–419.

    Article  MATH  Google Scholar 

  10. M. Sh. Birman and M. Z. Solomyak,Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalizations, inEstimates and Asymptotics for Discrete Spectra of Integral and Differential Equations (M. Sh. Birman, ed.), Amer. Math. Soc., Providence, Ri; Adv. Soviet Math.7 (1991), 1–55.

    Google Scholar 

  11. M. Sh. Birman and D. R. Yafaev,The spectral shift function. The work of M. G. Krein and its further development, St. Petersburg Math. J.4 (1993), 833–870.

    MathSciNet  Google Scholar 

  12. M. Sh. Birman and D. R. Yafaev,Spectral properties of the scattering matrix, St. Petersburg Math. J.4 (1993), 1055–1079.

    MathSciNet  Google Scholar 

  13. R. W. Carey,A unitary invariant for pairs of self-adjoint operators, J. Reine Angew. Math.283 (1976), 294–312.

    MathSciNet  Google Scholar 

  14. J. M. Combes, P. D. Hislop and E. Mourre,Spectral averaging, perturbation of singular spectrum, and localization, Trans. Amer. Math. Soc.348 (1996), 4883–4894.

    Article  MATH  MathSciNet  Google Scholar 

  15. L. de Branges,Perturbations of self-adjoint transformations, Amer. J. Math.84 (1962), 543–560.

    Article  MATH  MathSciNet  Google Scholar 

  16. J.-P. Eckmann and C.-A. Pillet,Zeta functions with Dirichlet and Neumann boundary conditions for exterior domains, Helv. Phys. Acta70 (1997), 44–65.

    MATH  MathSciNet  Google Scholar 

  17. R. Geisler, V. Kostrykin and R. Schrader,Concavity properties of Krein's spectral shift function, Rev. Math. Phys.7 (1995), 161–181.

    Article  MATH  MathSciNet  Google Scholar 

  18. F. Gesztesy and K. A. Makarov,Some applications of the spectral shift operator, inOperator Theory and its Applications (A. G. Ramm, P. N. Shivakumar and A. V. Strauss, eds.), Fields Institute Communication Series, Amer. Math. Soc., Providence, RI, to appear.

  19. F. Gesztesy, K. A. Makarov and S. N. Naboko,The spectral shift operator, inMathematical Results in Quantum Mechanics (J. Dittrich, P. Exner and M. Tater, eds.), Operator Theory: Advances and Applications, Vol. 108, Birkhäuser, Basel, 1999, pp. 59–90.

    Google Scholar 

  20. F. Gesztesy and B. Simon,PitThe xi function, Acta Math.176 (1996), 49–71.

    Article  MATH  MathSciNet  Google Scholar 

  21. F. Gesztesy and E. Tsekanovskii,On matrix-valued Herglotz functions, Math. Nachr., to appear.

  22. I. C. Gohberg and M. G. Krein,Introduction to the Theory of Linear Nonselfadjoint Operators, Amer. Math. Soc., Providence, RI, 1969.

    MATH  Google Scholar 

  23. H. R. Grümm,Two theorems about C p, Rep. Math. Phys.4 (1973), 211–215.

    Article  MATH  Google Scholar 

  24. N. J. Kalton,A note on pairs of projections, Bol. Soc. Mat. Mexicana (3)3 (1997), 309–311.

    MATH  MathSciNet  Google Scholar 

  25. T. Kato,Notes on projections and perturbation theory, Technical Report No. 9, University of California at Berkeley, 1955.

  26. T. Kato,Wave operators and similarity for some non-selfadjoint operators, Math. Ann.162 (1966), 258–279.

    Article  MATH  MathSciNet  Google Scholar 

  27. T. Kato,Monotonicity theorems in scattering theory, Hadronic J.1 (1978), 134–154.

    MATH  MathSciNet  Google Scholar 

  28. R. Konno and S. T. Kuroda,On the finiteness of perturbed spectra, J. Fac. Sci. Univ. Tokyo, Sect. I13 (1966), 55–63.

    MathSciNet  MATH  Google Scholar 

  29. V. Kostrykin and R. Schrader,Scattering theory approach to random Schrödinger operators in one dimension, Rev. Math. Phys.11 (1999), 187–242.

    Article  MATH  MathSciNet  Google Scholar 

  30. M. G. Krein,On the trace formula in perturbation theory, Mat. Sbornik N.S.33(75) (1953), 597–626 (Russian).

    MathSciNet  Google Scholar 

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

    Google Scholar 

  32. M. G. Krein,On certain new studies in the perturbation theory for self-adjoint operators, inM. G. Krein, Topics in Differential and Integral Equations and Operator Theory (I. Gohberg, ed.), Birkhäuser, Basel, 1983.

    Google Scholar 

  33. M. G. Krein,On perturbation determinants and a trace formula for certain classes of pairs of operators, Amer. Math. Soc. Transl. (2)145 (1989), 39–84.

    MathSciNet  Google Scholar 

  34. M. G. Krein and V. A. Javrjan,On spectral shift functions arising in perturbations of a positive operator, J. Operator Theory6 (1981), 155–191 (Russian).

    MathSciNet  Google Scholar 

  35. I. M. Lifshits,On a problem of perturbation theory, Uspekhi Mat. Nauk7, No. 1 (1952), 171–180.

    MATH  Google Scholar 

  36. I. M. Lifshits,Some problems of the dynamic theory of nonideal crystal lattices, Nuovo Cimento Suppl.3 (Ser. X) (1956), 716–734.

    Article  Google Scholar 

  37. W. Müller,Relative zeta functions, relative determinants and scattering theory, Comm. Math. Phys.192 (1998), 309–347.

    Article  MATH  MathSciNet  Google Scholar 

  38. S. N. Naboko,Boundary values of analytic operator functions with a positive imaginary part, J. Soviet Math.44 (1989), 786–795.

    Article  MATH  MathSciNet  Google Scholar 

  39. S. N. Naboko,Nontagential boundary values of operator-valued R-functions in a half-plane, Leningrad Math. J.1 (1990), 1255–1278.

    MATH  MathSciNet  Google Scholar 

  40. A. B. Pushnitski,Spectral shift function of the Schrödinger operator in the large coupling constant limit, Comm. Partial Differential Equations, to appear.

  41. A. B. Pushnitski,Integral estimates for the spectral shift function, Algebra i Analiz10 (1998), 198–233; to appear in St. Petersburg Math. J.10, no. 6, (1999).

    Google Scholar 

  42. M. Reed and B. Simon,Methods of Modern Mathematical Physics I, Functional Analysis, 2nd edn., Academic Press, New York, 1980.

    MATH  Google Scholar 

  43. D. Robert,Semi-classical approximation in quantum mechanics. A survey of old and recent mathematical results, Helv. Phys. Acta71 (1998), 44–116.

    MathSciNet  Google Scholar 

  44. O. L. Safronov,The discrete spectrum in a gap of the continuous spectrum for variable sign perturbations with large coupling constant, St. Petersburg Math. J.8 (1997), 307–331.

    MathSciNet  Google Scholar 

  45. J. Schwinger,On the bound states of a given potential, Proc. Nat. Acad. Sci. U.S.A.47 (1961), 122–129.

    Article  MathSciNet  Google Scholar 

  46. B. Simon,Trace Ideals and their Applications, Cambridge University Press, Cambridge, 1979.

    MATH  Google Scholar 

  47. B. Simon,Spectral analysis of rank one perturbations and applications, CRM Proc. Lecture Notes8 (1995), 109–149.

    Google Scholar 

  48. B. Simon,Spectral averaging and the Krein spectral shift, Proc. Amer. Math. Soc.126 (1998), 1409–1413.

    Article  MATH  MathSciNet  Google Scholar 

  49. K. B. Sinha,On the theorem of M. G. Krein, preprint, 1975, unpublished.

  50. A. V. Sobolev,Efficient bounds for the spectral shift function, Ann. Inst. H. PoincaréA58 (1993), 55–83.

    MathSciNet  Google Scholar 

  51. B. Sz.-Nagy and C. Foias,Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970.

    Google Scholar 

  52. D. R. Yafaev,Mathematical Scattering Theory, Amer. Math. Soc., Providence, RI, 1992.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Fritz Gesztesy & Konstantin A. Makarov

Authors
  1. Fritz Gesztesy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Konstantin A. Makarov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gesztesy, F., Makarov, K.A. The Ξ operator and its relation to Krein's spectral shift function. J. Anal. Math. 81, 139–183 (2000). https://doi.org/10.1007/BF02788988

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02788988

Keywords

Search

Navigation