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Spectral averaging and the Krein spectral shift

Author: Barry Simon
Journal: Proc. Amer. Math. Soc. 126 (1998), 1409-1413
MSC (1991): Primary 47B10, 47A60
MathSciNet review: 1443857
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Abstract: We provide a new proof of a theorem of Birman and Solomyak that if $A(s) = A_{0} + sB$ with $B\geq 0$ trace class and $d\mu _{s} (\cdot ) = \text{Tr}(B^{1/2} E_{A(s)}(\cdot ) B^{1/2})$, then $\int ^{1}_{0} [d\mu _{s} (\lambda )]\, ds = \xi (\lambda )\, d\lambda $, where $\xi $ is the Krein spectral shift from $A(0)$ to $A(1)$. Our main point is that this is a simple consequence of the formula $\frac{d}{ds} \text{Tr}(f(A(s))=\text{Tr}(Bf'(A(s)))$.

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  • 1. M. Š. Birman and M. Z. Solomjak, Remarks on the spectral shift function, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 27 (1972), 33–46 (Russian). Boundary value problems of mathematical physics and related questions in the theory of functions, 6. MR 0315482
  • 2. I. C. Gohberg and M. G. Kreĭn, Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Izdat. “Nauka”, Moscow, 1965 (Russian). MR 0220070
    I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. MR 0246142
  • 3. V. A. Javrjan, A certain inverse problem for Sturm-Liouville operators, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 6 (1971), no. 2–3, 246–251 (Russian, with Armenian and English summaries). MR 0301565
  • 4. S. Kotani, Lyapunov exponents and spectra for one-dimensional random Schrödinger operators, Random matrices and their applications (Brunswick, Maine, 1984) Contemp. Math., vol. 50, Amer. Math. Soc., Providence, RI, 1986, pp. 277–286. MR 841099, 10.1090/conm/050/841099
  • 5. M. G. Kreĭn, On the trace formula in perturbation theory, Mat. Sbornik N.S. 33(75) (1953), 597–626 (Russian). MR 0060742
  • 6. M. G. Kreĭn, On perturbation determinants and a trace formula for unitary and self-adjoint operators, Dokl. Akad. Nauk SSSR 144 (1962), 268–271 (Russian). MR 0139006
  • 7. S. S. Gershteĭn, L. I. Ponomarev, and T. P. Puzynina, A quasiclassical approximation in the two-center problem, Soviet Physics JETP 21 (1965), 418–425. MR 0184647
    M. G. Kreĭn, Topics in differential and integral equations and operator theory, Operator Theory: Advances and Applications, vol. 7, Birkhäuser Verlag, Basel, 1983. Edited by I. Gohberg; Translated from the Russian by A. Iacob. MR 815109
  • 8. Barry Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526. MR 670130, 10.1090/S0273-0979-1982-15041-8
  • 9. Barry Simon, Spectral analysis of rank one perturbations and applications, Mathematical quantum theory. II. Schrödinger operators (Vancouver, BC, 1993) CRM Proc. Lecture Notes, vol. 8, Amer. Math. Soc., Providence, RI, 1995, pp. 109–149. MR 1332038
  • 10. Franz Wegner, Bounds on the density of states in disordered systems, Z. Phys. B 44 (1981), no. 1-2, 9–15. MR 639135, 10.1007/BF01292646

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Additional Information

Barry Simon
Affiliation: Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, California 91125

Received by editor(s): October 14, 1996
Additional Notes: This material is based upon work supported by the National Science Foundation under Grant No. DMS-9401491. The government has certain rights in this material.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 Barry Simon