The Spectrum of Periodic Point Perturbations and the Krein Resolvent Formula

  • J. Brüning
  • V. A. Geyler
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 117)


We study periodic point perturbations H of a periodic elliptic operator H 0 on a connected complete non-compact Riemannian manifold X,endowed with an isometric, effective, properly discontinuous, and co-compact action of a discrete group Γ. Under some conditions H 0, we prove that the gaps of the spectrum σ (H) are labelled in a natural way by elements of the K 0-group of a certain C*-algebra. In particular, if the group Γ has the Kadison property then σ (H) has band structure. The Krein resolvent formula plays a crucial role in proving the main results.

AMS Classification

58625 81Q10 


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  1. 1.
    B.S. Pavlov, The theory of extensions and explicitly solvable models (in Russian). Uspekhi Mat. Nauk 42 no. 6 (1987), 99–131; Engl. trans].: Russ. Math. Surv. 42 no. 6 (1987), 127–168.zbMATHCrossRefGoogle Scholar
  2. 2.
    S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden, Solvable models in quantum mechanics. Springer-Verlag, Berlin etc., 1988.zbMATHCrossRefGoogle Scholar
  3. 3.
    R. Kronig and W.G. Penney, Quantum mechanics of electrons in crystal lattices. Proc. Roy. Soc. (London) 130A (1931), 499–513.Google Scholar
  4. 4.
    A. Grossmann, R. 110egh-Krohn and M. Mebkhout, The one-particle theory of periodic point interactions. Comm. Math. Phys. 77 (1080), 87–100.CrossRefGoogle Scholar
  5. 5.
    Yu.E. Karpeshina, Spectrum and eigenfunctions of Schrödinger operator with zero-range potential of the homogenous lattice type in three dimensional space (in Russian). Teor. i. Mat. Fiz. 57 (1983), 304–313; Engl. trans].: Theor. and Math. Phys. 57 (1983), 1156–1162.Google Scholar
  6. 6.
    S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden, Point interactions in two dimensions: Basic properties, approximations and applications to solid state physics. J. reine u. angew. Math. 380 (1987), 87–107.zbMATHGoogle Scholar
  7. 7.
    T. Sunada, Group C* -algebras and the spectrum of a periodic Schrodinger operator on a manifold. Can. J. Math. 44 (1992), 180–193.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    J. Brüning and T. Sunada, On the spectrum of periodic elliptic operators. Nagoya Math. J. 126 (1992), 159–171.zbMATHGoogle Scholar
  9. 9.
    J. Brüning and T. Sunada, On the spectrum of gauge-periodic elliptic operators. Astérisque 210 (1992), 65–74.Google Scholar
  10. 10.
    M.G. Krein and H.K. Langer, Defect subspace and generalized resolvents of an Hermitian operators in the spaceIIk (in Russian). Funk. Anal. i Prilozhen. 5, no. 2 (1971), 59–71; Engl. trans].: Funct. Anal. and its Appl. 5 (1971), 217–228.Google Scholar
  11. 11.
    J. Brüning and V. A. Geyler, Gauge periodic point perturbations on the Lobachevsky plane. Preprint SFB 288, Berlin, 1998.Google Scholar
  12. 12.
    T. Sunada, Euclidean versus non-euclidean aspects in spectral geometry. Progr. Theor. Phys. Suppl. 116 (1994), 235–250.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    N.E. Hurt, Quantum chaos and mesoscopic systems. Mathematical methods in the quantum signatures of chaos. Kluwer Ac. Publ., Dordrecht etc., 1997.zbMATHGoogle Scholar
  14. 14.
    M.L. Leadbeater, C.L. Foden and J. H. Burrougher, e.a. Magnetotransport in a non-planar two-dimensional electron gas. Phys. Rev. B. 52 (1995), 8629–8632.CrossRefGoogle Scholar
  15. 15.
    D.B. Efremov and M.A. Shubin, Spectral asymptotics of elliptic operators of the Schrödinger type on the Lobachevsky space (in Russian). Trudy Sem. I.G. Petrovsky 15 (1991), 3–32.MathSciNetGoogle Scholar
  16. 16.
    M.A. Shubin, Spectral theory of elliptic operators on non-compact manifolds. Astérisque 207 (1992), 35–108.MathSciNetGoogle Scholar
  17. 17.
    A.V. Bukhvalov, Applicatons of the method of the theory of order bounded operators in the L P -spaces (in Russian). Uspekhi Mat. Nauk 38 no. 6 (1983), 37–83.MathSciNetGoogle Scholar
  18. 18.
    V.B. Korotkov, Integral operators (in Russian). Nauka, Novosibirsk, 1983.Google Scholar
  19. 19.
    V.A. Geyler, V.A. Margulis and I.I. Chuchaev, Zero-range potentials and Carleman operators (in Russian). Sibir. Mat. Zhurn. 36 (1995), 828–841; Engl. trans].: Siberian Math. J. 36 (1995), 714–726.Google Scholar
  20. 20.
    P.R. Halmos and V.S. Sunder, Bounded linear operators on L 2 -spaces. SpringerVerlag, New York etc., 1978.CrossRefGoogle Scholar
  21. 21.
    M.A. Shubin, Pseudo-difference operators and their Green functions (in Russian). Izv. AN SSSR. Ser. Mat. 49 (1985), 652–67]; Engl. transl.: Math. USSR. Izvestiya. 26 (1986), 605–622.MathSciNetzbMATHGoogle Scholar
  22. 22.
    S.A. Gredeskul, M. Zusman, Y. Avishai and M.Ya. Azbel, Spectral properties and localization of an electron in a two-dimensional system with point scatterers in a magnetic field. Phys. Reps. 288 (1997), 223–257.CrossRefGoogle Scholar
  23. 23.
    V.A. Geyler, The two-dimensional Schrödinger operators with a uniform magnetic field and its perturbation by periodic zero-range potentials (in Russian). Algebra i Analiz, 3, no. 3 (1991), 1–48; Engl. transl.: St.-Petersburg Math. J. 3 (1992), 489–532.Google Scholar
  24. 24.
    S. Albeverio and V.A. Geyler, The band structure of the general periodic SchrOdinger operator with point interactions (to be published).Google Scholar
  25. 25.
    Yu.G. Shondin, Semibounded local Hamiltonians for perturbation of the Laplacian supported by curves with angle points in R 4 (in Russian). Teoret. i Mat. Fiz. 106 (1996), 179–199.MathSciNetGoogle Scholar
  26. 26.
    J. Bellissard, Gap labelling theorems for Schrödinger operators. In: From Number Theory to Physics. I Eds. Waldschmidt M. c.a. Springer-Verlag, Berlin etc., 1992, 538–630.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2000

Authors and Affiliations

  • J. Brüning
    • 1
  • V. A. Geyler
    • 2
  1. 1.Institute of MathematicsHumboldt University at BerlinBerlinGermany
  2. 2.Department of MathematicsMordovian State UniversitySaranskRussia

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