Letters in Mathematical Physics

, Volume 106, Issue 4, pp 485–507 | Cite as

The Density of Surface States as the Total Time Delay

  • Hermann Schulz-Baldes


For a scattering problem of tight-binding Bloch electrons by a weak random surface potential, a generalized Levinson theorem is put forward showing the equality of the total density of surface states and the density of the total time delay. The proof uses explicit formulas for the wave operators in the new rescaled energy and interaction representation, as well as an index theorem for adequate associated operator algebras.


surface scattering Levinson theorem 

Mathematics Subject Classification

81U99 47A40 19K56 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Amrein, W.O., Boutet de Monvel, A., Georgescu, V.: \({C_{0}}\)-groups, commutator methods and spectral theory of N-body Hamiltonians. Progress Math., vol. 135. Birkhäuser, Basel (1996)Google Scholar
  2. 2.
    Bellissard, J.: K-Theory of \({C^*}\)-algebras in solid state physics. In: Dorlas, T.C., Hugenholtz, M.N., Winnink, M. (eds.) Statistical Mechanics and Field Theory, Mathematical Aspects. Lecture Notes in Physics, vol. 257, pp. 99–156 (1986)Google Scholar
  3. 3.
    Bellissard J., Schulz-Baldes H.: Scattering theory for lattice operators in dimension \({d \geq 3}\). Rev. Math. Phys. 24, 1250020 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Blackadar B.: K-Theory for Operator Algebras, 2nd edn. Cambridge University Press, Cambridge(1998)MATHGoogle Scholar
  5. 5.
    Chahrour A.: Densité intégrée d’états surfaciques et fonction généralisée de déplacement spectral pour un opérateur de Schrödinger surfacique ergodique. Helv. Phys. Acta. 72, 93–122 (1999)MathSciNetGoogle Scholar
  6. 6.
    Chahrour A., Sahbani J.: On the spectral and scattering theory of the Schrödinger operator with surface potential. Rev. Math. Phys. 12, 561–574 (2000)MathSciNetMATHGoogle Scholar
  7. 7.
    Englisch H., Kirsch W., Schröder M., Simon B.: Random Hamiltonians ergodic in all but one direction. Commun. Math. Phys. 128, 613–625 (1990)MathSciNetCrossRefMATHADSGoogle Scholar
  8. 8.
    Georgescu, V., Iftimovici, A.: C\({^*}\)-algebras of quantum Hamiltonians. In: Operator Algebras and Mathematical Physics, Conference Proceedings 2001, 123–167, Theta Foundation (2003)Google Scholar
  9. 9.
    Graf G.-M., Schenker D.: 2-magnon scattering in the Heisenberg model. Ann. de l’Inst. H. Poincaré, Sect. A. 67, 91–107 (1997)MathSciNetMATHGoogle Scholar
  10. 10.
    Jaksic, V., Molchanov, S., Pastur, L.: On the propagation properties of surface waves. In: Papanicolaou, G. (ed.) Wave propagation in complex media. IMA Vol. Math. Appl. 96, 145–154Google Scholar
  11. 11.
    Jaksic V., Molchanov S.: On the surface spectrum in dimension two. Helv. Phys. Acta. 71, 629–657 (1999)MathSciNetMATHGoogle Scholar
  12. 12.
    Jaksic V., Molchanov S.: Localization of surface spectra. Commun. Math. Phys. 208, 153–172 (1999)MathSciNetCrossRefMATHADSGoogle Scholar
  13. 13.
    Jaksic V., Last Y.: Corrugated surfaces and the a.c. spectrum. Rev. Math. Phys. 12, 1465–1503 (2000)MathSciNetMATHGoogle Scholar
  14. 14.
    Jaksic V., Last Y.: Surface states and spectra. Commun. Math. Phys. 218, 459–477 (2001)MathSciNetCrossRefMATHADSGoogle Scholar
  15. 15.
    Kellendonk J., Richard S.: Levinson’s theorem for Schrödinger operators with point interaction: a topological approach. J. Phys. A 39, 14397–14403 (2006)MathSciNetCrossRefMATHADSGoogle Scholar
  16. 16.
    Kellendonk J., Richard S.: On the wave operators and Levinsons theorem for potential scattering in \({{\mathbb R}^3}\). Asian-Eur. J. Math. 5, 1250004 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kohmoto M., Koma T., Nakamura S.: The spectral shift function and the Friedel sum rule. Ann. H. Poincaré. 14, 1413–1424 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kirsch W., Klopp F.: The band-edge behavior of the density of surfacic states. Math. Phys. Anal. Geom. 8, 315–360 (2006)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kostrykin V., Schrader R.: Regularity of the surface density of states. J. Funct. Anal. 187, 227–246 (2001)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Levinson N.: On the uniqueness of the potential in a Schrödinger equation for a given asymptotic phase. Kgl. Danske Videnskabernes Selskab Mat.-fys. Medd. 25, 3–29 (1949)MathSciNetMATHGoogle Scholar
  21. 21.
    Richard S., Tiedrade Aldecoa R.: New expressions for the wave operators of Schrödinger operators in \({{\mathbb R}^3}\). Lett. Math. Phys. 103, 1207–1221 (2013)MathSciNetCrossRefMATHADSGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department MathematikUniversität Erlangen-NürnbergErlangenGermany

Personalised recommendations